Optimal Volume-Sensitive Bounds for Polytope Approximation
share
Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body K of diameter Δ in 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 ε. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that Θ((Δ/ε)^(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 volume diameter Δ_d to be the diameter of a Euclidean ball of the same volume as K, and define its surface diameter Δ_d-1 analogously for surface area. It follows from generalizations of the isoperimetric inequality that Δ≥Δ_d-1≥Δ_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((Δ_d-1/ε)^(d-1)/2). In this paper, we strengthen this by proving the existence of an approximation with O((Δ_d/ε)^(d-1)/2) facets.
READ FULL TEXT