On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds (Q2185013)

Given a convex body \(K\) in \(\mathbb{R}^n\), let \(\operatorname{vol}(K)\), \(D(K)\), and \(w(K)\) denote the volume, diameter, and the minimum width of \(K\), respectively. The main concern of the paper is studying sharp estimates and asymptotics of the corresponding isodiametric and isominwidth quotients of \(K\), which are defined by \[ \operatorname{idq} (K)=\frac{\operatorname{vol}(K)}{D(K)^n},\ \operatorname{iwq}(K)=\frac{\operatorname{vol}(K)}{w(K)^n}. \] In particular, the authors prove the following statements. Theorem. If \(K\) is in isodiametric position, i.e. \(\operatorname{idq} (K)=\max_{A \in GL_n(\mathbb{R})}(AK)\), then \[ \operatorname{idq}(K)\ge \frac{1}{\sqrt{n!}\, n^{n/2}}. \] If, moreover, \(K\) is origin-symmetric, then \(\operatorname{idq}(K)\ge 1/n!\), where the equality holds if and only if \(K\) is a regular crosspolytope. Theorem. There exists \(A \in GL_n (\mathbb{R})\) such that \(\operatorname{iwq}(AK)\le 1\), where the equality holds if and only if \(AK\) is a cube. The paper contains good introduction and description of the related tools, including results of Makai Jr, Ball, Behrend, Barthe, Dvoretzky-Rogers, and others.