Generalization of the Brunn–Minkowski inequality in the form of Hadwiger

A class of domain functionals has been built in the Euclidean space. The Brunn–Minkowski type of inequality has been applied to the said class and proved for it. Functional building has been performed using the point of minimum of function of n variables bound with functionals, proof of existence of which is the important part of the proposed research. We have introduced special cases of functionals for which the point of minimum can be found explicitly. The resulting Brunn–Minkowski type of inequality generalizes the corresponding inequality for moments of inertia in relation to the center of mass and hyperplanes proven by H. Hadwiger. It is worth mentioning that the point of minimum of functional in general case does not coincide with the center of mass. Coincidence occurs only in special cases, which is proven by the particular examples in this study.