The Brunn-Minkowski inequality for volume differences
Suppose that \(K\), \(L\), \(D\), \(D'\) are compact domains in \(\mathbb{R}^n\) such that \(D\) and \(D'\) are homothetic and convex and \(D\subset K\), \(D'\subset L\). It is proved (in a more general form) that for the volume \(V\) one has \[ ((V(K+ L)- V(D+ D'))^{1/n}\geq (V(K)- V(D))^{1/n}+ (V(L)- V(D'))^{1/n}. \] This extends the Brunn-Minkowski inequality, and it is deduced from the latter with the help of an algebraic inequality of Bellman. Similar extensions are given for Minkowski's inequality for mixed volumes, the matrix analogue of the Brunn-Minkowski inequality, and the generalized Brunn-Minkowski inequality for quermassintegrals. Reviewer's remark: The convexity of \(D\) and \(D'\) is not assumed in the paper, but without it the employed equation \(V(D+ D')^{1/n}= V(D)^{1/n}+ V(D')^{1/n}\) is not true in general. Moreover, the equality conditions stated in Theorem 1 do not hold without additional convexity assumptions.
scientific article; zbMATH DE number 2107254