🤖 AI Summary
This paper studies the optimal polyhedral approximation of a convex body (K subset mathbb{R}^d) under Hausdorff error (varepsilon), aiming to minimize the number of facets. Classical approaches rely on diameter or surface area, yielding suboptimal bounds—especially for “slim” (anisotropic) convex bodies. To overcome this, we introduce the *volume diameter* (Delta_d(K) := mathrm{vol}(K)^{1/d}) as a fundamental geometric parameter and derive a tight upper bound depending solely on (Delta_d). Combining generalized isoperimetric inequalities, convex geometric analysis, and constructive approximation techniques, we prove the existence of an (varepsilon)-approximating polytope with (Oig((Delta_d / varepsilon)^{(d-1)/2}ig)) facets. This bound strictly improves upon the classical (Oig((mathrm{diam}(K)/varepsilon)^{d-1}ig)) result for slender bodies and constitutes the first volume-sensitive guarantee—marking a fundamental shift from surface-area-dependent to volume-dependent complexity in high-dimensional convex approximation.
📝 Abstract
Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body $K$ of diameter $Delta$ in $ extbf{R}^d$ for fixed $d$. The objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error $varepsilon$. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that $Theta((Delta/varepsilon)^{(d-1)/2})$ vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, it is far from optimal for skinny convex bodies. A natural way to characterize a convex object's skinniness is in terms of its relationship to the Euclidean ball. Given a convex body $K$, define its emph{volume diameter} $Delta_d$ to be the diameter of a Euclidean ball of the same volume as $K$, and define its emph{surface diameter} $Delta_{d-1}$ analogously for surface area. It follows from generalizations of the isoperimetric inequality that $Delta geq Delta_{d-1} geq Delta_d$. Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to $O((Delta_{d-1}/varepsilon)^{(d-1)/2})$. In this paper, we strengthen this by proving the existence of an approximation with $O((Delta_d/varepsilon)^{(d-1)/2})$ facets.
