Abstract:
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$m=1$.
Keywords:Brunn–Minkowski inequality, Prékopa–Leindler inequality, concave function, convex body, power moments.
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