Boundary restricted Brunn-Minkowski inequalities (Q6592229)

The following Brunn-Minkowski type inequality is the main result of this paper. If \(K,T\subset {\mathbb R}^n\) (\(n\geq 2\)) are compact sets with connected boundaries, then \N\[\N\operatorname{vol}\left(\frac{\partial K+\partial T}{2}\right)\ge \sqrt{\operatorname{vol}(K)\operatorname{vol}(T)}.\N\]\NWhen \(K\) and \(T\) are convex bodies, equality holds if and only if either \(K,T\) are translates, or \(n=2\) and and \(K,T\) are homothetic and centrally symmetric. The inequality is extended to \(n>2\) compact sets. Another generalization of the inequality says that, under the same assumptions and for \(\lambda\in (0,1)\), \N\[\N\operatorname{vol}(\lambda\partial K+(1-\lambda)\partial T)\operatorname{vol}(\lambda \partial T+(1-\lambda)\partial K) \ge\operatorname{vol}(K)\operatorname{vol}(T)\cdot(1-|1-2\lambda|^{n})^2.\N\]\NFurther it is proved that under suitable restrictions (depending on the dimension) on the volume ratio of \(K\) and \(T\) one has \N\[\N\operatorname{vol}(\partial K+\partial T)^{2/n} \ge\operatorname{vol}(K)^{2/n} + \operatorname{vol}(T)^{2/n}.\N\]\NAmong the proof tools are the geometric structure of boundary sums, the classical Brunn-Minkowski inequality, and for the last inequality a result of \textit{S. J. Szarek} and \textit{D. Voiculescu} [Commun. Math. Phys. 178, No. 3, 563--570 (1996; Zbl 0863.46042)].