Convex Sets with Large Distortion

Abstract

Given a compact convex body \({\mathcal{L}}\) in a Euclidean vector space \({\mathcal{E}}\) with a fixed base point \({\mathcal{O}}\) in the interior of \({\mathcal{L}}\), an affine invariant \(\sigma({\mathcal{L}})\) can be defined that measures how distorted \({\mathcal{L}}\) is with respect to \({\mathcal{O}}\). The two extreme values of \(\sigma({\mathcal{L}})\) are 1 corresponding to a simplex, and \(( {\rm dim} {\mathcal{E}} + 1)/2\) corresponding to a (centrally) symmetric \({\mathcal{L}}\). In this paper we study the structure of \({\mathcal{L}}\) when \(\sigma({\mathcal{L}}) < 1+1/(1+dim{\mathcal{E}})\). We construct a polytope that contains \({\mathcal{L}}\), study its combinatorial structure, and prove that \({\mathcal{L}}\) is between two simplices scaled in the ratio \(1/(2 +{\rm dim} \mathcal {E} - (1 +{\ rm dim} {\mathcal{E}}) \sigma({\mathcal{L}}) \div 1\). This, in turn, gives an upper bound on the volume of \({\mathcal{L}}\) in terms of \(\sigma({\mathcal{L}})\) and the inscribed simplex.