![Page 1: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/1.jpg)
What is the perimeter of a log–concave function?
Andrea Colesanti, in collaboration with Ilaria Fragala
Convexity and asymptotic geometric analysis
Centre de Recherches MathematiquesMontreal, 16–20 April 2012
![Page 2: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/2.jpg)
Log–concave functions
We consider functions f : Rn → R+ of the form
f = e−u ,
whereu : Rn → (−∞,∞] is convex.
(with u 6≡ ∞). We will always assume
lim|x |→∞
f (x) = 0 ⇔ lim|x |→∞
u(x) =∞ .
Example.
f (x) = χK (x) =
{1 if x ∈ K ,0 if x 6∈ K ,
where K ⊂ Rn is a convex body. In this case
u(x) = IK (x) =
{0 if x ∈ K ,∞ if x 6∈ K ,
![Page 3: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/3.jpg)
Log–concave functionsWe consider functions f : Rn → R+ of the form
f = e−u ,
whereu : Rn → (−∞,∞] is convex.
(with u 6≡ ∞).
We will always assume
lim|x |→∞
f (x) = 0 ⇔ lim|x |→∞
u(x) =∞ .
Example.
f (x) = χK (x) =
{1 if x ∈ K ,0 if x 6∈ K ,
where K ⊂ Rn is a convex body. In this case
u(x) = IK (x) =
{0 if x ∈ K ,∞ if x 6∈ K ,
![Page 4: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/4.jpg)
Log–concave functionsWe consider functions f : Rn → R+ of the form
f = e−u ,
whereu : Rn → (−∞,∞] is convex.
(with u 6≡ ∞). We will always assume
lim|x |→∞
f (x) = 0
⇔ lim|x |→∞
u(x) =∞ .
Example.
f (x) = χK (x) =
{1 if x ∈ K ,0 if x 6∈ K ,
where K ⊂ Rn is a convex body. In this case
u(x) = IK (x) =
{0 if x ∈ K ,∞ if x 6∈ K ,
![Page 5: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/5.jpg)
Log–concave functionsWe consider functions f : Rn → R+ of the form
f = e−u ,
whereu : Rn → (−∞,∞] is convex.
(with u 6≡ ∞). We will always assume
lim|x |→∞
f (x) = 0 ⇔ lim|x |→∞
u(x) =∞ .
Example.
f (x) = χK (x) =
{1 if x ∈ K ,0 if x 6∈ K ,
where K ⊂ Rn is a convex body. In this case
u(x) = IK (x) =
{0 if x ∈ K ,∞ if x 6∈ K ,
![Page 6: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/6.jpg)
Log–concave functionsWe consider functions f : Rn → R+ of the form
f = e−u ,
whereu : Rn → (−∞,∞] is convex.
(with u 6≡ ∞). We will always assume
lim|x |→∞
f (x) = 0 ⇔ lim|x |→∞
u(x) =∞ .
Example.
f (x) = χK (x) =
{1 if x ∈ K ,0 if x 6∈ K ,
where K ⊂ Rn is a convex body. In this case
u(x) = IK (x) =
{0 if x ∈ K ,∞ if x 6∈ K ,
![Page 7: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/7.jpg)
Log–concave functionsWe consider functions f : Rn → R+ of the form
f = e−u ,
whereu : Rn → (−∞,∞] is convex.
(with u 6≡ ∞). We will always assume
lim|x |→∞
f (x) = 0 ⇔ lim|x |→∞
u(x) =∞ .
Example.
f (x) = χK (x) =
{1 if x ∈ K ,0 if x 6∈ K ,
where K ⊂ Rn is a convex body.
In this case
u(x) = IK (x) =
{0 if x ∈ K ,∞ if x 6∈ K ,
![Page 8: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/8.jpg)
Log–concave functionsWe consider functions f : Rn → R+ of the form
f = e−u ,
whereu : Rn → (−∞,∞] is convex.
(with u 6≡ ∞). We will always assume
lim|x |→∞
f (x) = 0 ⇔ lim|x |→∞
u(x) =∞ .
Example.
f (x) = χK (x) =
{1 if x ∈ K ,0 if x 6∈ K ,
where K ⊂ Rn is a convex body. In this case
u(x) = IK (x) =
{0 if x ∈ K ,∞ if x 6∈ K ,
![Page 9: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/9.jpg)
Operations on log–concave functions
Let f and g be log–concave and s > 0. We set
(f ⊕ g)(z) := supx+y=z
f (x)g(y) ,
(s · f )(x) := f s(x
s
).
I These operations preserve log–concavity: f ⊕ g and s · f arelog–concave.
I If K and L are convex bodies and t, s > 0, then
s · χK ⊕ t · χL = χsK+tL ,
wheresK + tL = {sx + ty : x ∈ K , y ∈ L} .
![Page 10: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/10.jpg)
Operations on log–concave functions
Let f and g be log–concave and s > 0.
We set
(f ⊕ g)(z) := supx+y=z
f (x)g(y) ,
(s · f )(x) := f s(x
s
).
I These operations preserve log–concavity: f ⊕ g and s · f arelog–concave.
I If K and L are convex bodies and t, s > 0, then
s · χK ⊕ t · χL = χsK+tL ,
wheresK + tL = {sx + ty : x ∈ K , y ∈ L} .
![Page 11: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/11.jpg)
Operations on log–concave functions
Let f and g be log–concave and s > 0. We set
(f ⊕ g)(z) := supx+y=z
f (x)g(y) ,
(s · f )(x) := f s(x
s
).
I These operations preserve log–concavity: f ⊕ g and s · f arelog–concave.
I If K and L are convex bodies and t, s > 0, then
s · χK ⊕ t · χL = χsK+tL ,
wheresK + tL = {sx + ty : x ∈ K , y ∈ L} .
![Page 12: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/12.jpg)
Operations on log–concave functions
Let f and g be log–concave and s > 0. We set
(f ⊕ g)(z) := supx+y=z
f (x)g(y) ,
(s · f )(x) := f s(x
s
).
I These operations preserve log–concavity: f ⊕ g and s · f arelog–concave.
I If K and L are convex bodies and t, s > 0, then
s · χK ⊕ t · χL = χsK+tL ,
wheresK + tL = {sx + ty : x ∈ K , y ∈ L} .
![Page 13: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/13.jpg)
Operations on log–concave functions
Let f and g be log–concave and s > 0. We set
(f ⊕ g)(z) := supx+y=z
f (x)g(y) ,
(s · f )(x) := f s(x
s
).
I These operations preserve log–concavity: f ⊕ g and s · f arelog–concave.
I If K and L are convex bodies and t, s > 0, then
s · χK ⊕ t · χL = χsK+tL ,
wheresK + tL = {sx + ty : x ∈ K , y ∈ L} .
![Page 14: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/14.jpg)
Operations on log–concave functions
Let f and g be log–concave and s > 0. We set
(f ⊕ g)(z) := supx+y=z
f (x)g(y) ,
(s · f )(x) := f s(x
s
).
I These operations preserve log–concavity: f ⊕ g and s · f arelog–concave.
I If K and L are convex bodies and t, s > 0, then
s · χK ⊕ t · χL = χsK+tL ,
wheresK + tL = {sx + ty : x ∈ K , y ∈ L} .
![Page 15: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/15.jpg)
The integral functional and its derivative
For f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 16: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/16.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx
∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 17: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/17.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 18: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/18.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 19: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/19.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 20: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/20.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies)
then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 21: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/21.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε
= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 22: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/22.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 23: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/23.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ)
=
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 24: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/24.jpg)
The integral functional and its derivativeFor f log–concave (verifying the decay condition at infinity), weconsider
I (f ) =
∫Rn
f (x) dx ∈ [0,∞) .
I If f = χK (K convex body), I (f ) = V (K ) (the volume of K ).
We are interested in the limit
δI (f , g) := limε→0+
I (f ⊕ ε · g)− I (f )
ε.
I If f = χK and g = χL (K and L convex bodies) then
δI (f , g) = limε→0+
V (K + εL)− V (K )
ε= n V (K , . . . ,K , L)
=
∫Sn−1
hL(ξ) dσK (ξ) =
∫∂K
hL(νK (x)) dHn−1(x) .
![Page 25: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/25.jpg)
Representation formulas for δI (f , g) (1)
Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 26: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/26.jpg)
Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 27: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/27.jpg)
Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 28: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/28.jpg)
Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 29: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/29.jpg)
Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 30: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/30.jpg)
Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 31: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/31.jpg)
Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 32: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/32.jpg)
Representation formulas for δI (f , g) (1)Let f = e−u, g = e−v . The equality
δI (f , g) =
∫Rn
v∗(∇u(x)) f (x) dx ,
where v∗ is the usual conjugate of v :
v∗(y) = supx∈Rn
x · y − v(x) ,
has been proved in some special cases.
I Klartag & Milman (2005), Rotem (2011): f = density of theGaussian measure.
I C. & Fragala (2011): general case, but with several additionalassumptions concerning smoothness, strict convexity, decay atinfinity of f and g .
I see also: Klartag (2007), who proved analogous results in thecontext of p–concave functions.
None of them includes the case: f = χK , g = χL, K and L convexbodies.
![Page 33: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/33.jpg)
Representation formulas for δI (f , g) (2)
Let f = e−u and g = e−v and set
Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .
Sf and Sg are convex sets; let Hf and Hg be their supportfunctions. Then the equality
δI (f , g) =
∫Sf
v∗(∇u(x)) f (x) dx +
∫∂Sf
Hg (ν(x)) f (x) dHn−1(x) ,
where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)
![Page 34: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/34.jpg)
Representation formulas for δI (f , g) (2)
Let f = e−u and g = e−v and set
Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .
Sf and Sg are convex sets; let Hf and Hg be their supportfunctions. Then the equality
δI (f , g) =
∫Sf
v∗(∇u(x)) f (x) dx +
∫∂Sf
Hg (ν(x)) f (x) dHn−1(x) ,
where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)
![Page 35: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/35.jpg)
Representation formulas for δI (f , g) (2)
Let f = e−u and g = e−v and set
Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .
Sf and Sg are convex sets; let Hf and Hg be their supportfunctions.
Then the equality
δI (f , g) =
∫Sf
v∗(∇u(x)) f (x) dx +
∫∂Sf
Hg (ν(x)) f (x) dHn−1(x) ,
where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)
![Page 36: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/36.jpg)
Representation formulas for δI (f , g) (2)
Let f = e−u and g = e−v and set
Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .
Sf and Sg are convex sets; let Hf and Hg be their supportfunctions. Then the equality
δI (f , g) =
∫Sf
v∗(∇u(x)) f (x) dx +
∫∂Sf
Hg (ν(x)) f (x) dHn−1(x) ,
where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)
![Page 37: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/37.jpg)
Representation formulas for δI (f , g) (2)
Let f = e−u and g = e−v and set
Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .
Sf and Sg are convex sets; let Hf and Hg be their supportfunctions. Then the equality
δI (f , g) =
∫Sf
v∗(∇u(x)) f (x) dx +
∫∂Sf
Hg (ν(x)) f (x) dHn−1(x) ,
where ν is the outer unit normal to Hf ,
has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)
![Page 38: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/38.jpg)
Representation formulas for δI (f , g) (2)
Let f = e−u and g = e−v and set
Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .
Sf and Sg are convex sets; let Hf and Hg be their supportfunctions. Then the equality
δI (f , g) =
∫Sf
v∗(∇u(x)) f (x) dx +
∫∂Sf
Hg (ν(x)) f (x) dHn−1(x) ,
where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .
(C. & Fragala, 2011)
![Page 39: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/39.jpg)
Representation formulas for δI (f , g) (2)
Let f = e−u and g = e−v and set
Sf = {x ∈ Rn : f (x) > 0} , Sg = {x ∈ Rn : g(x) > 0} .
Sf and Sg are convex sets; let Hf and Hg be their supportfunctions. Then the equality
δI (f , g) =
∫Sf
v∗(∇u(x)) f (x) dx +
∫∂Sf
Hg (ν(x)) f (x) dHn−1(x) ,
where ν is the outer unit normal to Hf , has been proved underseveral additional assumptions on f , g , Sf and Sg .(C. & Fragala, 2011)
![Page 40: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/40.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f , as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 41: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/41.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f , as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 42: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/42.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f , as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 43: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/43.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f , as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 44: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/44.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f ,
as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 45: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/45.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f , as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 46: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/46.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f , as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 47: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/47.jpg)
Perimeter
If K ⊂ Rn is a convex body, its perimeter, i.e. Hn−1(∂K ), can bedefined as
P(K ) = limε→0+
V (K + εB)− V (K )
ε,
where B is the unit ball of Rn.
I Define a corresponding notion of perimeter of a log-concavefunction f , as
P(f ) = limε→0+
I (f ⊕ ε · g)− I (f )
ε= δI (f , g) ,
with an appropriate choice of a unit ball g in the space oflog-concave functions.
Question: is there a natural candidate for g?
![Page 48: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/48.jpg)
The unit ball in the class of log–concave functions
I A natural possibility is to chose
g(x) = γn(x) = e−|x |2/2 ;
this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].
I A drastically different choice is
g = χB = characteristic function of the unit ball of Rn,
which is what we consider here.
I There are clearly many other choices, (e.g.
g(x) = e−|x |p/p , p ≥ 1 .)
![Page 49: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/49.jpg)
The unit ball in the class of log–concave functions
I A natural possibility is to chose
g(x) = γn(x) = e−|x |2/2 ;
this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].
I A drastically different choice is
g = χB = characteristic function of the unit ball of Rn,
which is what we consider here.
I There are clearly many other choices, (e.g.
g(x) = e−|x |p/p , p ≥ 1 .)
![Page 50: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/50.jpg)
The unit ball in the class of log–concave functions
I A natural possibility is to chose
g(x) = γn(x) = e−|x |2/2 ;
this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].
I A drastically different choice is
g = χB = characteristic function of the unit ball of Rn,
which is what we consider here.
I There are clearly many other choices, (e.g.
g(x) = e−|x |p/p , p ≥ 1 .)
![Page 51: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/51.jpg)
The unit ball in the class of log–concave functions
I A natural possibility is to chose
g(x) = γn(x) = e−|x |2/2 ;
this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].
I A drastically different choice is
g = χB = characteristic function of the unit ball of Rn,
which is what we consider here.
I There are clearly many other choices, (e.g.
g(x) = e−|x |p/p , p ≥ 1 .)
![Page 52: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/52.jpg)
The unit ball in the class of log–concave functions
I A natural possibility is to chose
g(x) = γn(x) = e−|x |2/2 ;
this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].
I A drastically different choice is
g = χB = characteristic function of the unit ball of Rn,
which is what we consider here.
I There are clearly many other choices, (e.g.
g(x) = e−|x |p/p , p ≥ 1 .)
![Page 53: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/53.jpg)
The unit ball in the class of log–concave functions
I A natural possibility is to chose
g(x) = γn(x) = e−|x |2/2 ;
this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].
I A drastically different choice is
g = χB = characteristic function of the unit ball of Rn,
which is what we consider here.
I There are clearly many other choices,
(e.g.
g(x) = e−|x |p/p , p ≥ 1 .)
![Page 54: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/54.jpg)
The unit ball in the class of log–concave functions
I A natural possibility is to chose
g(x) = γn(x) = e−|x |2/2 ;
this choice was made also by Klartag & Milman and Rotem intheir definition of mean width of a log–concave function, andin [C. & Fragala, 2012].
I A drastically different choice is
g = χB = characteristic function of the unit ball of Rn,
which is what we consider here.
I There are clearly many other choices, (e.g.
g(x) = e−|x |p/p , p ≥ 1 .)
![Page 55: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/55.jpg)
Definition of P(f ) and a useful formula
Let B be the unit ball of Rn. For a log–concave function f = e−u
define:
P(f ) := δI (f , χB) = limε→0+
I (f ⊕ ε · χB)− I (f )
ε.
Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε. Moreover, its (right) derivative atε = 0, i.e. P(f ), is
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
Note that {f > t} is convex for every t, by the log–concavity of f .
![Page 56: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/56.jpg)
Definition of P(f ) and a useful formula
Let B be the unit ball of Rn.
For a log–concave function f = e−u
define:
P(f ) := δI (f , χB) = limε→0+
I (f ⊕ ε · χB)− I (f )
ε.
Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε. Moreover, its (right) derivative atε = 0, i.e. P(f ), is
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
Note that {f > t} is convex for every t, by the log–concavity of f .
![Page 57: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/57.jpg)
Definition of P(f ) and a useful formula
Let B be the unit ball of Rn. For a log–concave function f = e−u
define:
P(f ) := δI (f , χB) = limε→0+
I (f ⊕ ε · χB)− I (f )
ε.
Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε. Moreover, its (right) derivative atε = 0, i.e. P(f ), is
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
Note that {f > t} is convex for every t, by the log–concavity of f .
![Page 58: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/58.jpg)
Definition of P(f ) and a useful formula
Let B be the unit ball of Rn. For a log–concave function f = e−u
define:
P(f ) := δI (f , χB) = limε→0+
I (f ⊕ ε · χB)− I (f )
ε.
Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε.
Moreover, its (right) derivative atε = 0, i.e. P(f ), is
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
Note that {f > t} is convex for every t, by the log–concavity of f .
![Page 59: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/59.jpg)
Definition of P(f ) and a useful formula
Let B be the unit ball of Rn. For a log–concave function f = e−u
define:
P(f ) := δI (f , χB) = limε→0+
I (f ⊕ ε · χB)− I (f )
ε.
Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε. Moreover, its (right) derivative atε = 0, i.e. P(f ), is
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
Note that {f > t} is convex for every t, by the log–concavity of f .
![Page 60: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/60.jpg)
Definition of P(f ) and a useful formula
Let B be the unit ball of Rn. For a log–concave function f = e−u
define:
P(f ) := δI (f , χB) = limε→0+
I (f ⊕ ε · χB)− I (f )
ε.
Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε. Moreover, its (right) derivative atε = 0, i.e. P(f ), is
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
Note that {f > t} is convex for every t, by the log–concavity of f .
![Page 61: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/61.jpg)
Definition of P(f ) and a useful formula
Let B be the unit ball of Rn. For a log–concave function f = e−u
define:
P(f ) := δI (f , χB) = limε→0+
I (f ⊕ ε · χB)− I (f )
ε.
Thm. (C. & Fragala, 2012) The function ε→ I (f ⊕ ε · χB) is apolynomial of degree n w.r.t. ε. Moreover, its (right) derivative atε = 0, i.e. P(f ), is
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
Note that {f > t} is convex for every t, by the log–concavity of f .
![Page 62: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/62.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 63: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/63.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave.
By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 64: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/64.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula,
if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 65: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/65.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 66: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/66.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 67: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/67.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn)
= total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 68: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/68.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 69: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/69.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 70: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/70.jpg)
Perimeter and total variation
Let f = e−u : Rn → R be log–concave. By the previous result andcoarea formula, if moreover f ∈ C 1(Rn), we have
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt
=
∫Rn
|∇f | dx .
In the general case
P(f ) = |Df |(Rn) = total variation of f in Rn
=
∫Sf
|∇f | dx +
∫∂Sf
f dHn−1 .
These equalities confirm the validity of the representation formulasfor δI (f , g) seen before when f is arbitrary and g = χB .
![Page 71: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/71.jpg)
Properties of the perimeter
Recall:
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
I Monotonicity:f ≤ g ⇒ P(f ) ≤ P(g) .
I Valuation property: for every f and g log–concave such thatf ∧ g is log–concave:
P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .
I Brunn-Minkowski inequality: for every f and g log–concave
P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .
This result was proved by Bobkov (2007).
![Page 72: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/72.jpg)
Properties of the perimeterRecall:
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
I Monotonicity:f ≤ g ⇒ P(f ) ≤ P(g) .
I Valuation property: for every f and g log–concave such thatf ∧ g is log–concave:
P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .
I Brunn-Minkowski inequality: for every f and g log–concave
P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .
This result was proved by Bobkov (2007).
![Page 73: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/73.jpg)
Properties of the perimeterRecall:
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
I Monotonicity:f ≤ g ⇒ P(f ) ≤ P(g) .
I Valuation property: for every f and g log–concave such thatf ∧ g is log–concave:
P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .
I Brunn-Minkowski inequality: for every f and g log–concave
P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .
This result was proved by Bobkov (2007).
![Page 74: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/74.jpg)
Properties of the perimeterRecall:
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
I Monotonicity:f ≤ g ⇒ P(f ) ≤ P(g) .
I Valuation property: for every f and g log–concave such thatf ∧ g is log–concave:
P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .
I Brunn-Minkowski inequality: for every f and g log–concave
P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .
This result was proved by Bobkov (2007).
![Page 75: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/75.jpg)
Properties of the perimeterRecall:
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
I Monotonicity:f ≤ g ⇒ P(f ) ≤ P(g) .
I Valuation property: for every f and g log–concave such thatf ∧ g is log–concave:
P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .
I Brunn-Minkowski inequality: for every f and g log–concave
P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .
This result was proved by Bobkov (2007).
![Page 76: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/76.jpg)
Properties of the perimeterRecall:
P(f ) =
∫ ∞0
[perimeter of the level set {f > t}] dt .
I Monotonicity:f ≤ g ⇒ P(f ) ≤ P(g) .
I Valuation property: for every f and g log–concave such thatf ∧ g is log–concave:
P(f ∧ g) + P(f ∨ g) = P(f ) + P(g) .
I Brunn-Minkowski inequality: for every f and g log–concave
P((1− t) · f ⊕ t · g) ≥ (P(f ))1−t (P(g))t .
This result was proved by Bobkov (2007).
![Page 77: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/77.jpg)
A Cauchy type formula
Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥. We define theprojection of f onto Eξ as
fξ(x′) := sup
t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .
Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R. Let
In−1(fξ) :=
∫Eξ
fξ(x′) dx ′ .
Then
P(f ) = c(n)
∫Sn−1
In−1(fξ) dHn−1(ξ) .
![Page 78: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/78.jpg)
A Cauchy type formula
Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥.
We define theprojection of f onto Eξ as
fξ(x′) := sup
t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .
Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R. Let
In−1(fξ) :=
∫Eξ
fξ(x′) dx ′ .
Then
P(f ) = c(n)
∫Sn−1
In−1(fξ) dHn−1(ξ) .
![Page 79: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/79.jpg)
A Cauchy type formula
Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥. We define theprojection of f onto Eξ as
fξ(x′) := sup
t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .
Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R. Let
In−1(fξ) :=
∫Eξ
fξ(x′) dx ′ .
Then
P(f ) = c(n)
∫Sn−1
In−1(fξ) dHn−1(ξ) .
![Page 80: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/80.jpg)
A Cauchy type formula
Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥. We define theprojection of f onto Eξ as
fξ(x′) := sup
t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .
Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R.
Let
In−1(fξ) :=
∫Eξ
fξ(x′) dx ′ .
Then
P(f ) = c(n)
∫Sn−1
In−1(fξ) dHn−1(ξ) .
![Page 81: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/81.jpg)
A Cauchy type formula
Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥. We define theprojection of f onto Eξ as
fξ(x′) := sup
t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .
Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R. Let
In−1(fξ) :=
∫Eξ
fξ(x′) dx ′ .
Then
P(f ) = c(n)
∫Sn−1
In−1(fξ) dHn−1(ξ) .
![Page 82: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/82.jpg)
A Cauchy type formula
Let f be log–concave, ξ ∈ Sn−1 and Eξ = ξ⊥. We define theprojection of f onto Eξ as
fξ(x′) := sup
t∈Rf (x ′ + tξ) , x ′ ∈ Eξ .
Put simply, the graph of fξ is the projection of the graph of f ontoEξ × R. Let
In−1(fξ) :=
∫Eξ
fξ(x′) dx ′ .
Then
P(f ) = c(n)
∫Sn−1
In−1(fξ) dHn−1(ξ) .
![Page 83: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/83.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).
![Page 84: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/84.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).
![Page 85: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/85.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).
![Page 86: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/86.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).
![Page 87: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/87.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).
![Page 88: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/88.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).
![Page 89: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/89.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball.
Equality holds ifff = χB (up to translation).
![Page 90: Convexity and asymptotic geometric analysisweb.math.unifi.it/users/colesant/ricerca/Montreal2012.pdf · Convexity and asymptotic geometric analysis Centre de Recherches Mathematiques](https://reader030.vdocuments.us/reader030/viewer/2022041100/5ed8388f0fa3e705ec0e0f95/html5/thumbnails/90.jpg)
Isoperimetric inequalities
I Using the definition of P(f ) and Prekopa–Leindler inequalitywe find:
P(f ) ≥ c(n)I (f ) + Ent(f ) , (1)
where: Ent(f ) =
∫Rn
f log(f ) dx − I (f ) log(I (f )) ,
and equality holds iff f = χB (up to a translation).
I The Sobolev inequality in Rn for functions of boundedvariation yelds
P(f ) ≥ nω1/nn ‖f ‖ n
n−1, (2)
where ωn is the volume of the unit ball. Equality holds ifff = χB (up to translation).