This document discusses open questions about the asymptotic geometry of convex sets in hyperbolic space, including: 1) Whether there is a hyperbolic analogue of Dvoretzky's theorem on sections of convex bodies being almost spherical. 2) What convex bodies in hyperbolic space look like, and whether there is a relationship between the "set at infinity" where a body intersects the ideal boundary and its volume. 3) The author proves that for any proper convex set in hyperbolic space, the dimension of its limit set at infinity is bounded above by (n-1)/2.

1. Asymptotic
Geometry of Convex
Sets
{ Igor Rivin (Temple and IAS)

2.  Itai Benjamini and Gadi Kosma had asked
whether there is a hyperbolic analogue of
Dvoretzky’s Theorem
The motivating question

3.  (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

4.  What would a hyperbolic Dvoretzky theorem
say?
Hyperbolic Dvoretzky?

5.  Simpler question: what do convex bodies in
hyperbolic space look like?
Hyperbolic Dvoretzky

6. Hyperbolic plane

7. Hyperbolic Space

8. A counterexample?

9. Is the volume finite?

10.  Is there any relationship between the “set
at infinity” and its volume?
The next problem

11.  If the body intersects the ideal
boundary in an open set, it has
infinite volume!
Obvious answer: YES

12.  (does not help us with our
example…)
Not quite satisfying

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

14.  We call a (convex) set proper, if
its volume is positive and
finite.
Finally, a definition…

15.  The limit set C∞ of a convex set
C is the intersection of C with
the ideal boundary of Hn.
And another…

16.  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

17. 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

18.  Finite area if (and only if) the limit set C∞ is
finite.
Dimension 2

19.  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

20.  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