Brunn--Minkowski type inequality for generalized power moments in the form of Hadwiger

B. S. Timergaliev

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2016), pp. 92-107

Résumé

In this paper we built a class of domain functionals in Euclidian space and proved Brunn–Minkowski type inequality applied to the mentioned class. The resulting inequality generalizes corresponding inequality for moments of inertia in relation to the center of mass and hyperplanes proven by H. Hadwiger. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Define the functional $$ I(k;m;\Omega)=\int\limits_{\Omega}\left(\alpha_{1}|x_{1}-s_{1}|^{k}+\cdots+\alpha_{n}|x_{n}-s_{n}|^{k}\right)^{m}dx, $$ where $k\in(0,1]$ at $m\in(0,1)\cup(1,+\infty)$ and $k\in(0,+\infty)$ at $m=1$; $\alpha_{j}(j=\overline{1,n})\in(0,+\infty)$—arbitrary real numbers, $s_{1},s_{2},\ldots,s_{n}$—coordinates of the minimum point of the function $$ \begin{aligned} I(y)=\int\limits_{\Omega} \left(\alpha_{1}|x_{1}-y_{1}|^{k}+\alpha_{2}|x_{2}-y_{2}|^{k}\,+\cdots\right.\\ \left. +\cdots\,+\alpha_{n}|x_{n}-y_{n}|^{k}\right)^{m}dx, \ dx=dx_{1}dx_{2}\cdots dx_{n} \end{aligned} $$ of the variables $y=(y_{1},y_{2},\ldots,y_{n})\in \mathbb{R}^{n}$, where $x_{1},x_{2},\ldots,x_{n}$—Cartesian coordinates of the point $x\in\Omega$. The main result of this paper is the following Theorem. Let $\Omega_{0}, \Omega_{1}$ be a bounded domains in $\mathbb{R}^{n}$, that can be represented as the the union of a finite number of convex domains. Then the functional $I(k;m;\Omega)^{1/(km+n)}$ concave: $$ I(k;m;\Omega_{t})^{1/(km+n)}\geq(1-t)I(k;m;\Omega_{0})^{1/(km+n)}+tI(k;m;\Omega_{1})^{1/(km+n)}, $$ where $\Omega_{t}=\{(1-t)z_{0}+tz_{1} \ | \ z_{0}\in\Omega_{0}, z_{1}\in\Omega_{1}\}$, $0\leq t\leq 1$, $k\in(0,1]$ at $m\in(0,1)\cup(1,+\infty)$ and $k\in(0,+\infty)$ at $m=1$.

Détail

Keywords: Brunn–Minkowski inequality, Prékopa–Leindler inequality, concave function, convex body, power moments.