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Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces

Steven Hoehner, Carsten Schütt, Elisabeth Werner

TL;DR

We address the problem of approximating the $d$-dimensional Euclidean ball $B_d$ by polytopes with a fixed number of $k$-faces for $k\in\{0,1,\dots,d-1\}$, extending Gruber’s question beyond the classical inscribed/circumscribed models. Our approach blends combinatorial polytope theory, polar duality, and isoperimetric-type inequalities to derive sharp nonasymptotic lower bounds for intrinsic-volume differences $\delta_j$ and the Hausdorff distance, and to establish asymptotic rates for best-approximation problems. We extend prior results (BHK, LSW, HSW) to non-simplicial polytopes and to broader $k$-face ranges, and we introduce and exploit the total intrinsic-volume metric $\delta_\Sigma$ to obtain comprehensive multivariate lower bounds, including a dimension-dependent improvement in arbitrarily positioned settings. A key technical tool is the Dirichlet–Voronoi tiling number $\operatorname{div}_{d-1}$, for which we provide a simplified proof of its asymptotics and connect it to lower bounds via the Wills functional. Our findings yield near-optimal order results in several regimes and offer a concrete framework for extending lower bounds to remaining $k$-ranges and to Hausdorff-type metrics in broader convex-body classes.

Abstract

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and the Hausdorff metric. In the case of inscribed and circumscribed polytopes, our main results extend the previously obtained bounds from $k=0$ and $k=d-1$, respectively, to half of the $f$-vector of the approximating polytope. For arbitrarily positioned polytopes, we also improve a special case of a result of K. J. Böröczky ({\it J. Approx. Theory}, 2000) by a factor of dimension. This paper addresses a question of P. M. Gruber ({\it Convex and Discrete Geometry}, p. 216), who asked for results on the approximation of convex bodies by polytopes with a fixed number of $k$-faces when $1\leq k\leq d-2$.

Approximation of the Euclidean ball by polytopes with a fixed number of $k$-faces

TL;DR

We address the problem of approximating the -dimensional Euclidean ball by polytopes with a fixed number of -faces for , extending Gruber’s question beyond the classical inscribed/circumscribed models. Our approach blends combinatorial polytope theory, polar duality, and isoperimetric-type inequalities to derive sharp nonasymptotic lower bounds for intrinsic-volume differences and the Hausdorff distance, and to establish asymptotic rates for best-approximation problems. We extend prior results (BHK, LSW, HSW) to non-simplicial polytopes and to broader -face ranges, and we introduce and exploit the total intrinsic-volume metric to obtain comprehensive multivariate lower bounds, including a dimension-dependent improvement in arbitrarily positioned settings. A key technical tool is the Dirichlet–Voronoi tiling number , for which we provide a simplified proof of its asymptotics and connect it to lower bounds via the Wills functional. Our findings yield near-optimal order results in several regimes and offer a concrete framework for extending lower bounds to remaining -ranges and to Hausdorff-type metrics in broader convex-body classes.

Abstract

We derive lower estimates for the approximation of the -dimensional Euclidean ball by polytopes with a fixed number of -dimensional faces, . The metrics considered include the intrinsic volume difference and the Hausdorff metric. In the case of inscribed and circumscribed polytopes, our main results extend the previously obtained bounds from and , respectively, to half of the -vector of the approximating polytope. For arbitrarily positioned polytopes, we also improve a special case of a result of K. J. Böröczky ({\it J. Approx. Theory}, 2000) by a factor of dimension. This paper addresses a question of P. M. Gruber ({\it Convex and Discrete Geometry}, p. 216), who asked for results on the approximation of convex bodies by polytopes with a fixed number of -faces when .
Paper Structure (27 sections, 23 theorems, 110 equations)

This paper contains 27 sections, 23 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Overview of the paper
  3. Background and notation
  4. Main Results
  5. General lower bounds
  6. Inequalities for asymptotic best approximation
  7. The total intrinsic volume metric
  8. Arbitrarily positioned polytopes
  9. Symmetric volume difference
  10. Surface area deviation
  11. Proofs of Theorems \ref{['inscribed-mw-cor-kfaces']} and \ref{['circumscribed-vol-cor-kfaces']}
  12. Proof of Theorem \ref{['inscribed-mw-cor-kfaces']}
  13. Proof of Theorem \ref{['circumscribed-vol-cor-kfaces']}
  14. Asymptotic inequalities
  15. On the Dirichlet--Voronoi tiling number $\operatorname{div}_{d-1}$
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $d\geq 2$ be an integer and let $k\in\{0,1,\ldots,\lfloor\frac{d}{2}\rfloor\}$. For all polytopes $P_M\subset B_d$ with at most $M$$k$-faces and which contain $o$ in their interiors, we have

Theorems & Definitions (44)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • proof
  • ...and 34 more