One polytope fits all: Characterization of the Euclidean ball via simultaneous intrinsic volume approximation

TL;DR

This paper characterizes the Euclidean ball as the unique convex body that can be asymptotically optimally approximated by inscribed or circumscribed polytopes in terms of intrinsic volumes, using deterministic and probabilistic models.

Contribution

It establishes the rigidity of the Euclidean ball as the only shape allowing simultaneous optimal approximation of all intrinsic volumes in both deterministic and probabilistic settings.

Findings

Euclidean ball uniquely approximable for all intrinsic volumes asymptotically

Single sampling density yields optimal approximation only for Euclidean ball

Dual results for circumscribed polytopes using polarity

Abstract

We investigate the asymptotic best approximation of a smooth, strictly convex body $K$ in $\mathbb{R}^d$ by inscribed polytopes with a restricted number of vertices under the intrinsic volume difference. We prove rigidity phenomena in both the deterministic and probabilistic settings. In the deterministic model of inscribed approximation, we show that if a single sequence of polytopes is asymptotically best for the volume and mean width difference simultaneously, then $K$ must be a Euclidean ball. In particular, the Euclidean ball is the unique $C_+^2$ convex body for which one sequence of polytopes can approximate all intrinsic volumes simultaneously at the optimal asymptotic rate. In the probabilistic model, we prove a stronger statement: if a single sampling density on $\partial K$ yields random inscribed polytopes that are asymptotically optimal (in expectation) for any two distinct intrinsic volume deviations, then $K$ must be a Euclidean ball. Moreover, using polarity, we establish dual versions of this rigidity theorem for polytopes circumscribed about $K$ (with a restricted number of facets) in the volume and mean width cases, again in both deterministic and probabilistic frameworks. The proofs use tools from asymptotic quantization theory together with the curvature-based optimal vertex distributions. These results resolve an open question posed by Besau, Hoehner and Kur ({\it IMRN}, 2021).