The Brunn-Minkowski-type inequality (Q553437)

For star bodies \(K,L\) in \({\mathbb R}^n\) (\(n >2\)) and for \(p<-1\), the inequality \[ \tilde V_p({\mathbf I}^\circ (K\tilde + L))^{-1/p(n-1)} \leq \tilde V_p({\mathbf I}^\circ K)^{-1/p(n-1)} + \tilde V_p({\mathbf I}^\circ L)^{-1/p(n-1)} \] is proved, with equality if and only if \(K\) and \(L\) are dilates. Here \(\tilde +\) denotes radial addition, \textbf{I} is the intersection body operator, \textbf{I}\(^\circ K\) and \textbf{I}\( K\) have reciprocal radial functions, and \(\tilde V_p\) is the dual quermassintegral of order \(p\). The proof involves several applications of Hölder's and Minkowski's inequalities for integrals.