The logarithmic Minkowski inequality for non-symmetric convex bodies
The classical Brunn-Minkowski inequality is a fundamental inequality in the Brunn-Minkowski theory. This inequality is proved to be a very strong tool In [\textit{E. Lutwak}, J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007)] the classical Brunn-Minkowski theory was extended to the \(L_{p}\) Brunn-Minkowski theory for all \(p>1\). Also, Lutwak established the corresponding \(L_{p}\) Brunn-Minkowski inequality. In the landmark paper [Adv. Math. 231, No. 3--4, 1974--1997 (2012; Zbl 1258.52005)] \textit{K. Böröczky jun.} et al., defined the log-Minkowski combination for convex bodies, posed the conjectured log-Brunn-Minkowski inequality (for origin symmetric convex bodies), and confirmed the log-Brunn-Minkowski inequality in \(\mathbb{R}^{2}\). Let \(K_{1}, K_{2}\) be two origin-symmetric convex bodies and \(0<\lambda<1\), then the conjectured log-Brunn-Minkowski inequality can be stated as the following: \[ V\big((1-\lambda)\cdot K_{1}+_{0}\lambda\cdot K_{2}\big)\geq V(K_{1})^{1-\lambda}V(K_{2})^{\lambda}, \] where \((1-\lambda)\cdot K_{1}+_{0}\lambda\cdot K_{2}\) is defined by the Wulff shape \[ (1-\lambda)\cdot K_{1}+_{0}\lambda\cdot K_{2}=\bigcap_{u\in S^{n-1}}\left\{x\in\mathbb{R}^{n}: x\cdot u\leq h_{K_{1}}(u)^{1-\lambda}h_{K_{2}}(u)^{\lambda} \right\}. \] In the paper under review (1) for some particular cases, the author validates the conjectured log-Minkowski problem, (2) the author proves some variants of the log-Minkowski inequality for non-symmetric convex bodies, (3) the author proves a dual type log-Minkowski inequality, (4) the author also find applications of (2), and establishes some new inequalities.
scientific article; zbMATH DE number 6523904
scientific article