Asymptotics of Convex sets in En and Hn

TL;DR

This paper investigates the geometric properties of convex sets in hyperbolic and Euclidean spaces, establishing bounds on the dimensions of their intersections with boundaries and analyzing sections through points.

Contribution

It provides sharp bounds on the Minkowski and Hausdorff dimensions of boundary intersections and characterizes sections of convex sets in hyperbolic and Euclidean spaces.

Findings

Boundary intersections have Minkowski dimension at most (n-1)/2.

Existence of bounded sections through any point with radius bounds.

Asymptotic estimates for sections as dimension n grows.

Abstract

We study convex sets C of finite (but non-zero volume in Hn and En. We show that the intersection of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n-1)/2, and this bound is sharp. In the hyperbolic case we show that for any k <= (n-1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of convex body in En, and give asymptotic estimates as 1 << k << n.