Isoperimetric problems for convex bodies and a localization lemma
The isoperimetric coefficient of a convex body \(K \subseteq \mathbb{R}^ n\) is defined as the largest number \(\psi= \psi(K)\) such that for every measurable subset \(S \subseteq K\) for which \(\partial_ K S\) (the boundary of \(S\), relative to \(K\)) has an \((n-1)\)-dimensional Minkowski measure, \[ \text{vol}_{n-1} (\partial_ K S)\geq \psi {{\text{vol} (S)\cdot \text{vol} (K\setminus S)} \over {\text{vol} (K)}}. \] Since 1989 several attempts have been made to find a sharp lower bound of the isoperimetric coefficient; the best result was obtained by Dyer and Frieze in 1992 when they gave a lower bound of \(\psi (K)\) in terms of the diameter. However, the diameter may not be the best possible measure to use in this problem; in its application to volume algorithms and related questions, these bounds are needed in the case that the body is rather ``round'' and so the bounds in terms of the diameter may not be sharp. The main result of this paper gives the following lower bound for \(\psi (K)\): \[ \psi (K)\geq {{\ln 2} \over {M_ 1 (K)}}, \tag{1} \] where \(M_ 1 (K)\) denotes the average distance of a point in \(K\) from its center of gravity. The main tool to obtain this result is a variant of a general ``Localization Lemma'' that reduces integral inequalities over the \(n\)- dimensional space to integral inequalities in a single variable. The first version of this ``Localization Lemma'' was proved by two of the authors (Lovász and Simonovits) in 1993. Another lower bound for \(\psi (K)\) is also proved in this paper: \[ \psi (K)\geq {1\over {\chi(K)}}, \tag{2} \] where \(\chi(K)= {1\over {\text{vol} (K)}} \int_ K \chi(x) dx\), and \(\chi (x)\) denotes the length of the longest segment contained in \(K\) with midpoint \(x\). The two lower bounds (1) and (2) are not comparable. At the end of the paper there is also given an upper bound for \(\psi (K)\): \[ \psi (K)\leq {{10} \over {\sqrt {\alpha (K)}}}, \tag{3} \] where \(\alpha (K)\) is the largest eigenvalue of the matrix of inertia of \(K\). The authors conjecture that this upper bound is identical, up to a constant factor, to \(\psi (K)\).
scientific article; zbMATH DE number 764243