Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Steven Hoehner

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).

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

Abstract

We investigate the asymptotic best approximation of a smooth, strictly convex body in 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 must be a Euclidean ball. In particular, the Euclidean ball is the unique 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 yields random inscribed polytopes that are asymptotically optimal (in expectation) for any two distinct intrinsic volume deviations, then must be a Euclidean ball. Moreover, using polarity, we establish dual versions of this rigidity theorem for polytopes circumscribed about (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).
Paper Structure (15 sections, 19 theorems, 116 equations)

This paper contains 15 sections, 19 theorems, 116 equations.

Table of Contents

  1. Introduction and main results
  2. Deterministic inscribed approximation and rigidity
  3. Asymptotically best sequences
  4. Asymptotic formulas for volume and mean width approximation
  5. Limiting vertex distributions for asymptotically best sequences
  6. Deterministic rigidity theorem for inscribed polytopes
  7. Random rigidity for inscribed polytopes
  8. Sharp asymptotics and optimal densities for random approximation
  9. Random rigidity for inscribed polytopes
  10. Dual rigidity for circumscribed polytopes
  11. Polarity and circumscribed vs. inscribed approximation
  12. Deterministic circumscribed approximation: setup and functionals
  13. Asymptotically best circumscribed sequences and limiting facet distributions
  14. Deterministic dual rigidity via polarity
  15. Random dual rigidity for circumscribed polytopes

Key Result

Theorem 1.1

Let $K\subset\mathbb{R}^d$ be a $C_+^2$ convex body. Suppose there exists a sequence of inscribed polytopes $\{P_N\}\subset\mathscr{P}_N(K)$ such that $P_N$ is an asymptotically best-approximating polytope for both $\delta_V$ and $\delta_W$. Then $K$ must have constant Gauss curvature, i.e., $K$ is

Theorems & Definitions (45)

  • Theorem 1.1: Deterministic rigidity for inscribed polytopes
  • Theorem 1.2: Random rigidity for inscribed polytopes
  • Definition 2.1: Asymptotically best approximants
  • Lemma 2.1: Asymptotically best sequences use asymptotically $N$ vertices
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4: Best inscribed volume approximation
  • Remark 2.5
  • Remark 2.6
  • ...and 35 more