Improvements of the analogy isoperimetric inequality and the theorem of stability of its extremal solution

Анотація

The following inequality is proved:
$V^{n/(n-m)}_m(A,B)-V^{m/(n-m)}(B)V(A)\ge (V^{1/(n-m)}_m(A,B)-$ $\rho V^{1/(n-m)}(B))^n - V^{m/(n-m)}(B)V(A_{-\rho}(B)), 0\le \rho\le q$,
his consequents and the theorem of stability of solution $X$  equation $V^n_m(X,B)-V^m(B)V^{n-m}(X)=0$ at $V(X)=V(B)$. In given inequality $V(A)$, $V(B)$ – the volumes of convex bodies $A$ and $B$ in $\mathbb{R}^n$ $n\ge 2$, $V_m(A,B)$ – $m$-mixed volume of bodies $A$  and $B$, $1\le m\le n-1$, $q$ – a capacity coeficient of $B$ in a body $A$, $A_{-\rho}(B)$) – internal body which is parallel to body $A$ relatively to $B$ with coeficient $\rho$. 

Mathematical Subject Classification: 52A38, 52A40.