The dual \(\phi\)-Brunn-Minkowski inequality
Let \(K, L\subset\mathbb{R}^n\) be star bodies (i.e., star-shaped sets with respect to the origin, having continuous radial function), and let \(\rho_K\) and \(\rho_L\) be the radial functions of \(K\) and \(L\), respectively. Let \(\phi\in C^1((0,\infty))\) be strictly decreasing and concave, and assume that \(\log\circ\phi^{-1}\) is a convex function. For \(\xi\in[0,1]\), the authors define the function \(\rho_\xi\colon\mathbb{S}^{n-1}\to\mathbb{R}\) as follows: \[ \rho_\xi(u)=\sup\left\{\tau>0\colon (1-\xi)\phi\left(\frac{\rho_K(u)}{\tau}\right)+\xi\phi\left(\frac{\rho_L(u)}{\tau}\right)\le\phi(1)\right\}. \] The authors denote by \(\tilde Q_{\phi,\xi}\) the star body having \(\rho_\xi\) as radial function. The following Brunn-Minkowski-type inequality is proved: \[ (1-\xi)\phi\left(\frac{ |K|^{1/n}}{|\tilde Q_{\phi,\xi}|^{1/n}}\right)+ \xi\phi\left(\frac{|L|^{1/n}}{|\tilde Q_{\phi,\xi}|^{1/n}}\right)\le\phi(1), \] here \(|\cdot|\) denotes the volume, i.e., the Lebesgue measure in \(\mathbb{R}^n\). Moreover, when \(\xi\in(0,1)\), equality holds if and only if \(K\) and \(L\) are dilates of each other. The second main result of the paper is that the previous inequality is equivalent to the following Minkowski-type inequality: \[ \int_{\mathbb{S}^{n-1}}\phi\left(\frac{\rho_L(u)}{\rho_K(u)}\right)\, d\tilde V_K(u)\ge \phi\left(\frac{|L|^{1/n}}{|K|^{1/n}}\right), \] where \(\tilde V_K\) denotes the so-called \textit{dual cone-volume probability measure} of \(K\).
scientific article; zbMATH DE number 7397593
scientific article