Duality Bounds for Convexified Packing in Hilbert Geometry

Abstract

Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$, where $0 \in \text{int} G$ and $G \subset \text{int} K$. Given $α> 0$, the Hilbert packing number $M_H(G, K; α)$ is the maximum cardinality of a set of points in $G$, each pair of which is separated by a distance of at least $α$ in the Hilbert geometry defined by $K$. The Hilbert convexified packing number $\widehat{M}_H(G, K; α)$ is the maximum length of a sequence of points in $G$, such that each point is separated by distance at least $α$ from the convex hull of its predecessors. We prove a dimension-free primal-polar bound for convexified packing in Hilbert geometry. Letting $G^\circ$ and $K^\circ$ denote the polar bodies, we show that there exist absolute constants $C, c > 0$ such that, for every $α> 0$, \[ \widehat{M}_H(G, K; α) ~ \leq ~ C \cdot \widehat{M}_H(K^\circ, G^\circ; cα)^2 M_H(K^\circ, G^\circ; cα). \] As a direct corollary, we have \[ \widehat{M}_H(G, K; α) ~ \leq ~ C \cdot M_H(K^\circ, G^\circ; cα)^3. \] Thus, the primal convexified packing number is bounded by a fixed polynomial in the ordinary packing number for the reversed polar bodies, with absolute constants independent of the dimension. This is motivated by the duality conjecture for packing and covering numbers, which relates covering $G$ by $K$ to covering $K^\circ$ by $G^\circ$. Our results represent a key step in extending the work of Artstein, Milman, Szarek, and Tomczak-Jaegermann from normed spaces to Hilbert geometries.