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Asymptotic Geometry of Convex Sets

Igor Rivin (Temple and IAS)

Itai Benjamini and Gadi Kosma had asked whether there is a hyperbolic analogue of Dvoretzky’s Theorem

The motivating question

(A. Dvoretzky, 1961): For any centrally symmetric convex body in RN there exists an almost spherical central section (this is often stated in terms of Banach spaces), almost spherical meaning that there is an inscribed and a circumscribed ball, with ratio of radii close to 1.

Dvoretzky’s theorem

What would a hyperbolic Dvoretzky theorem say?

Hyperbolic Dvoretzky?

Simpler question: what do convex bodies in hyperbolic space look like?

Hyperbolic Dvoretzky

Hyperbolic plane

Hyperbolic Space

A counterexample?

Is the volume finite?

Is there any relationship between the “set at infinity” and its volume?

The next problem

If the body intersects the ideal boundary in an open set, it has infinite volume!

Obvious answer: YES

(does not help us with our example…)

Not quite satisfying

Hyperplane in Hn intersects infinity in a set of codimension 1, has 0 volume. Not interesting…

Stupid example

We call a (convex) set proper, if its volume is positive and finite.

Finally, a definition…

The limit set C∞ of a convex set C is the intersection of C with the ideal boundary of Hn.

And another…

For any proper convex set in Hn, dim C∞ ≤(n-1)/2, Where the dimension is the upper

Minkowski dimension (which upper-bounds the Hausdorff dimension).

And a Theorem

If C∞ is smooth, then the volume of the convex hull C of C∞ is not greater than the floor of

n/2-1.

And another

Finite area if (and only if) the limit set C∞ is finite.

Dimension 2

The exist subsets of the two-sphere of arbitrary Hausdorff dimension smaller than 1, such that the volume of their convex hull is finite (based on generalized Sierpinski gaskets).

Open question: can you do dimension equal to 1?

Dimension 3

Fixing dimension not exceeding the “critical value” ((n-1)/2), one can always find a plane of that dimension such that the intersection of the plane and the body is bounded in terms of the volume and the dimension, and the inradius.

Non-asymptotic consequences