21.3 Affine dimension and relative interior_

Let corn (not nee. affine).To "upgrade"C to affine set, wediefireitsaffine hull

in

affcc)=}ï 1X,+... +Okxk1×1, "-1 NiceF.oi=1}

|_

coï toeg

c

as Cc)

_ a |....|in o

C

!_ Oaffil)

ï an=1R2

ï

ï!ï ï\I..

> after is smallest affine set cont!-dining C.

_

formal:_

t affihesets SHR.I

C s...aft)(E_

ï |Affine dimensionof cis the dimension of its off CCJ.

20

ego dim is1 20 dim is2.

Interior of." Set_

'

eg | ,in _

:... --

_ JoC

Mais

"all pts strictly in ï idle\1 _ |0 \\

interior is in ifY"_ |//

Xy zO ocean"interior is empty, sheetdir unused"is

Euclidean balls -: Box, in) =x

center XEIRr¢ radiusre0 11

{ }~ - y

ego |mm

x

yïï

xof ball in 2 DBlair)

Of

11*117 y +f T

Openclosed Sets -Li nit points: =Let c E IR"

|on noo I

egg cdodo£ it in ï _ inï

rin,fro

non

intersection

Formally:*ERR is a limitptof c if

t 470'Blast)n six)) (H(C ¢in

p setdiffeenceClosed set:_

A set Cis closed -ifc contains all its limitpts.

Open set: -N setC is open if it is notclosed.

ego |on Top M satL

ï

{ 0C=34 11)ï |_

(=[0,1,Indo

enclosed set

Formally:Throw iny "

missing limit-points,getclosure of_ --

C,denoted d (C).

old Old.00 ï .. a 6O

t Inerror: _ Int (c) ={ XEC] | |_ 33¢] 870 sit.Blx,r)E if

\

ï

hï

ïFig ! ~

ï

Boundary: -zS= cks)\int(s)in throw out interior

I

_ |4 ..... in in ---e.g. in ï ï 'aadd o S=(0,1]IR

Int(s):(017)ïS= alls), Int(s)= [o, Dto, 1)=10, 1}

|Tae inTaito5=10.0),IR?Ä note!Int(s):0-

265) =elses)\!.,!..ills) = [0.1]\ of [0, D.

Relative Interioran _ ` in

: Y interior relative tooff(s)"

rebirths)= Exespero st Blair)( )_} asraff(s)

r old def of interiort "brings invision down"e.g. |in oog

recall int (5=0an:

S

affls)relint(5)= (011)

Ex 2.2 S= ExctR3/-1fx,!1,-1×21 73=0}

^ |v

27itx

y

z

1|

-

_

] !mm2affls) =intcs)=relintls)=

215) =

{XEIR3/*3=0}of{xE1R3/-1<xll1-1<x24 X3=0}Eyelike)mailhikerd|} = 1, *3=0}