New geometric inequalities for hedgehogs (Q1300325)

A hypersurface \(H_h\) in Euclidean \((n+1)\)-space \(\mathbb{E}^{n+1}\) is called a hedgehog if it is the envelope of hyperplanes \(\langle x,p\rangle= h(p)\) where \(h\) is a function in \(\mathbb{C}^2(S^n,\mathbb{R})\), \(S^n\) the unit sphere in \(\mathbb{E}^{n+1}\). This is a generalization of the case where \(h\) is the support function of a convex body \(K\) and \(H_h\) coincides with the surface of \(K\). The author derives the Alexandrov-Fenchel inequality \[ \nu(K_1, \dots, K_{n +1}) \geq\nu (K_1,K_1, K_3,\dots, K_{n+1})\cdot \nu(K_2,K_2,K_3, \dots, K_{n+1}), \] where \(K_3,\dots, K_{n+1}\) denote ordinary convex bodies, only \(K_1\) and \(K_2\) are hedgehogs and \(\nu(\cdots)\) is a generalization of the mixed volume of convex bodies. Further inequalities for hedgehogs are derived from the one above. In particular, in \(\mathbb{E}^3\) a so far unknown inequality for a convex body \(K\) of the type \(V\geq f(S,M)\) is established, under a certain assumption on a hedgehog associated to the support function of \(K\), where \(V,S,M\) denote volume, surface area, integral of the mean curvature of \(K\), respectively. Brecause of the additional assumption, the conjecture of \textit{J. R. Sangwine-Yager} [Am. Math. Mon. 96, No. 3, 223-237 (1989; Zbl 0669.52008)] concerning the boundary of the Blaschke diagram for \(K\) is still open.