\(L_p\)-mixed intersection bodies and star duality
For star bodies \(K_1,\dots,K_n\) in \({\mathbb R}^n\), the authors introduce the \(q\)-dual mixed volume (\(q\not=0\)) by \[ \tilde V_q(K_1,\dots,K_n)=\omega_n\left(\frac{1}{n\omega_n} \int_{S^{n-1}} \rho^q(K_1,u)\cdots\rho^q(K_n,u)\, dS(u)\right)^{1/q}, \] where the notations are as usual (e.g., \(\rho\) is the radial function), and the \(L_p\)-mixed intersection body \({\mathbf I}_p(K_1, \dots, K_{n-1})\) (\(p<1\)) by \[ \rho^p({\mathbf I}_p(K_1,\dots,K_{n-1}),u) =\frac{2}{1-p}\tilde v_p(K_1\cap E_u,\dots,K_{n-1}E_u), \] where \(\tilde v_p\) is the \(p\)-dual mixed volume in \((n-1)\)-dimensional spaces. Various inequalities, involving these objects and the star duality \(K\mapsto K^\circ\), defined by \(\rho(K^\circ,\cdot)=1/\rho(K,\cdot)\), are obtained, mainly by applications of Hölder's integral inequality.
scientific article; zbMATH DE number 5826106
scientific article