Convex Approximation and the Hilbert Geometry

2026-07-30

AI Summary

This work addresses the problem of efficient approximate membership testing for high-dimensional convex bodies. It proposes an intrinsically geometric approach based on the Hilbert metric induced by the convex body, constructing a Delone set via Macbeath regions and ellipsoidal coverings, and enabling fast queries through ray-shooting. The main contributions include the first application of volume entropy to analyze the size of Delone sets, yielding tight bounds, and the design of a pointer-search mechanism guided by Hilbert distance, which effectively controls query time and significantly enhances efficiency in high-dimensional settings.

Abstract

The efficient representation of convex bodies in multi-dimensional spaces is a fundamental problem in computational geometry. Several key developments were recently brought about using a number of constructions utilizing Macbeath regions. In this paper, we present a novel intrinsic approach for approximate membership testing, where we carry out the entire development based on structures derived from the Hilbert metric associated with a convex body $K$ in $\mathbb{R}^d$. First, we revisit the construction of economical Delone sets, deriving the size bound based on the notion of volume entropy. Second, we design a new query structure based on a simple covering by ellipsoids, where queries are answered by ray shooting. As an added bonus, the intrinsic viewpoint facilitates finger searching, where the query time can be bounded by the distance traveled in the Hilbert metric.

Problem

• convex bodies
• approximate membership testing
• Hilbert metric
• computational geometry
• high-dimensional representation

Innovation

• Hilbert geometry
• convex approximation
• membership testing
• Delone sets
• ray shooting