Some inequalities about mixed volumes (Q1424034)

The authors obtain several inequalities about the quermassintegrals \(V_k(K)\) of a convex body \(K\) in the euclidean \(n\)-dimensional space. There is a close relationship between inequalities about quermassintegrals of convex bodies and inequalities about symmetric functions of positive reals or determinants of symmetric matrices. For instance, an inequality of Bergstrom asserts that if \(A\) and \(B\) are symmetric positive definite matrices and \(A_i\), \(B_i\) denote the submatrices obtained by deleting the \(i\)-th row and column, then \[ \frac{\det(A+B)}{\det(A_i+B_i)}\geq \frac{\det(A)}{\det(A_i)}+\frac{\det(B)}{\det(B_i)}. \] Milman asked if there is a version of Bergstrom's inequality in the theory of mixed volumes. An analogous question was asked by \textit{A. Dembo, T. M. Cover} and \textit{J. A. Thomas} [IEEE Trans. Inf. Theory 37, No. 6, 1501--1518 (1991; Zbl 0741.94001)] when they were looking for the dual of the Fisher information inequality. The authors provide an answer to both questions proving that the inequality \[ \frac{V_k(K+L)} {V_{k-1}(K+L)}\geq \frac{V_k(K)}{V_{k-1}(K)}+\frac{V_k(L)}{V_{k-1}(L)} \] holds for every pair of convex bodies \(K\) and \(L\) in the \(n\)-dimensional euclidean space if and only if \(k=2\) or \(k=1\). They also obtain several inequalities comparing the mixed volumes of convex bodies with the mixed volumes of their projections.