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

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 Ω be a bounded domain in R𝑛. Define the functional 𝐼(𝑘;𝑚; Ω) = w Ω (︀ 1|𝑥1 - 𝑠1|𝑘 + · · · + 𝑛|𝑥𝑛 - 𝑠𝑛|𝑘)︀𝑚 𝑑𝑥, where ∈ (0, 1] at ∈ (0, 1) ∪ (1,+∞) and ∈ (0,+∞) at = 1; 𝑗(𝑗 = = 1, 𝑛) ∈ (0,+∞) - arbitrary real numbers, 𝑠1, 𝑠2,..., - coordinates of the minimum point of the function 𝐼(𝑦) = w Ω (︀ 1|𝑥1 - 𝑦1|𝑘 + 2|𝑥2 - 𝑦2|𝑘 + · · · +· · · + 𝑛|𝑥𝑛 - 𝑦𝑛|𝑘)︀𝑚 𝑑𝑥, = 𝑑𝑥1𝑑𝑥2 · · · of the variables = (𝑦1, 𝑦2,..., 𝑦𝑛) ∈ R𝑛, where 𝑥1, 𝑥2,..., - Cartesian coordinates of the point ∈ Ω. The main result of this paper is the following Theorem. Let Ω0,Ω1 be a bounded domains in R𝑛, that can be represented as the the union of a finite number of convex domains. Then the functional 𝐼(𝑘;𝑚; Ω)1/(𝑘𝑚+𝑛) concave: 𝐼(𝑘;𝑚;Ω𝑡)1/(𝑘𝑚+𝑛) ≥ (1 - 𝑡)𝐼(𝑘;𝑚;Ω0)1/(𝑘𝑚+𝑛) + 𝑡𝐼(𝑘;𝑚;Ω1)1/(𝑘𝑚+𝑛), where Ω𝑡 = {(1 - 𝑡)𝑧0 + 𝑡𝑧1 | 𝑧0 ∈ Ω0, 𝑧1 ∈ Ω