Convex Hulls in the Hyperbolic Space

TL;DR

This paper establishes bounds on the volume of convex hulls in hyperbolic space, showing they are linearly related to the number of points and the volume of neighborhoods, with constants depending on the dimension.

Contribution

It proves universal volume bounds for convex hulls in hyperbolic space and relates convex hull volume to neighborhood volumes with explicit constants.

Findings

Convex hull volume is bounded by a universal constant times the number of points.

For each dimension, convex hull volume is controlled by the volume of a 1-neighborhood.

These bounds are dimension-dependent and provide new geometric insights.

Abstract

We show that there exists a universal constant C>0 such that the convex hull of any N points in the hyperbolic space H^n is of volume smaller than C N, and that for any dimension n there exists a constant C_n > 0 such that for any subset A of H^n, Vol(Conv(A_1)) < C_n Vol(A_1) where A_1 is the set of points of hyperbolic distance to A smaller than 1.