Brunn-Minkowski inequality for mixed intersection bodies
In 1985 Lutwak introduced the notion of mixed projection bodies and obtained the Brunn-Minkowski inequality for these bodies. In this paper the authors prove the corresponding inequality for mixed intersection bodies. Besides its intrinsic interest this result is also an interesting example showing the duality between projection and intersection bodies. In particular they prove the following results: Let \(x_1 \tilde + \cdots \tilde + x_r\) denote the usual vector sum of \(x_1, \dots , x_r\) if all \(x_1, \dots , x_r\) lie in the same \(1\)-dimensional subspace, and \(0\) otherwise. If \(K_1=\dots =K_{n-i}=K\) and \(K_{n-i+1}=\dots =K_n=L\) the dual mixed volumes are written as \(V_i(K,L)\). Let \(\widetilde W_i(K)\) denote the dual mixed volume \(\widetilde V_i(K,B)\). Let \(IK\) denote the intersection body of \(K\), and \(I_iK\) the \(i\)th mixed intersection body of \(K\). If \(K, L\) are \(n\)-dimensional star-shaped bodies, then for \(0 \leq i < n\), \[ \widetilde W_i(I(K\tilde + L))^{\frac1{(n-i)(n-1)}}\leq \widetilde W_i (IK)^{\frac1{(n-i)(n-1)}} + \widetilde W_i (IL)^{\frac1{(n-i)(n-1)}}, \] with equality if and only if \(K\) and \(L\) are dilates. They also obtain the following more general statement: If \(K, L\) are \(n\)-dimensional star-shaped bodies, then for \(0 \leq i < n\), and for \(0 \leq j <n-2\), \[ \widetilde W_i(I_j(K\tilde + L))^{\frac1{(n-i)(n-j-1)}}\leq \widetilde W_i (I_jK)^{\frac1{(n-i)(n-j-1)}} + \widetilde W_i (I_jL)^{\frac1{(n-i)(n-j-1)}}, \] with equality if and only if \(K\) and \(L\) are dilates.
scientific article; zbMATH DE number 2131162
scientific article