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

TL;DR

This paper establishes lower bounds for approximating Euclidean balls with polytopes having a fixed number of k-faces, extending previous bounds and improving known results in convex approximation theory.

Contribution

It extends existing bounds to polytopes with a fixed number of k-faces and improves approximation results for arbitrarily positioned polytopes.

Findings

Derived lower estimates for approximation metrics

Extended bounds from vertices and facets to intermediate faces

Improved a special case of a known approximation result

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. B\"or\"oczky ({\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 $1\leq k\leq d-2$.