Convex Approximation and the Hilbert Geometry

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.