A geometric inequality on mixed volumes
For convex bodies \(K,M\) in \(\mathbb{R}^n\), the inequality \[ V_{n- 1}(K,M)\leq \rho^{n-1}W_{n-1}(K) \tag{\(*\)} \] (where \(V_{n-1}(K,M)= V(K,M,\dots,M)\) denotes the mixed volume, \(W_{n-1}(K)\) the \((n-1)\)st quermassintegral, and \(\rho\) is the circumradius of \(M\)) follows directly from the monotonicity of mixed volumes. The author extends \((*)\) to quasiconvex sets \(W\) (these are compact sets with smooth boundary which have convex hyperplane sections in a certain direction). The extension states that \[ V_{n-1}(c(W),M)\leq {\textstyle {\frac1n}} \rho^{n-1}M_{n-2}(W), \] where \(c(W)\) is the convex hull and \(M_{n- 2}(W)\) the \((n-2)\)th curvature integral. The proof is based on a connectivity result for quasi-convex sets and on the principal kinematic formula.
scientific article; zbMATH DE number 955340