On a degeneracy ratio for bounded convex polytopes

TL;DR

This paper introduces a measure of the roundness of convex polytopes and relates it to the smallest singular value of the matrix of their facet normals, providing a mathematical link between geometric shape and linear algebra.

Contribution

It establishes an upper bound on the roundness measure of convex polytopes based on the smallest singular value of the normals matrix, offering a new geometric-algebraic connection.

Findings

Derived a bound linking roundness and singular values

Provided a new perspective on polytope shape analysis

Enhanced understanding of convex polytope geometry

Abstract

We consider a quantity that measures the roundness of a bounded, convex $d$-polytope in $\mathbb{R}^d$. We majorise this quantity in terms of the smallest singular value of the matrix of outer unit normals to the facets of the polytope.