Stability of Minkowski and Brunn's equations solutions

V. I. Diskant

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 6 (1999) no. 3, pp. 245-252

Résumé

The following theorem of stability of Minkowski and Brunn's equations solutions are proved. Theorem 1. If $$ V_1^n(A, X)-V(X)V^{n-1}(A)<\varepsilon,\ \ 0\leq\varepsilon<\varepsilon_0,\ \ V(X)=V(sA),\ \ s>0, $$ then $\delta(sA, X)<C\varepsilon^{1/n}$. Theorem 2. If $$ V^{1/n}(H_{\frac{1}{2}})-\frac{1}{2}V^{1/n}(A)-\frac{1}{2}V^{1/n}(X)<\varepsilon,\ \ 0\leq\varepsilon<\varepsilon_0,\ \ V(X)=V(sA),\ \ s>0, $$ then $\delta(sA, X)<C\varepsilon^{1/n}$. In these theorems $A$ and $X$ — convex bodies in $R^n$, $V(A)$ — volume $A$, $V_1(A, X)$ — the first mixed volume $A$ and $X$, $H_{\frac{1}{2}}=\frac{1}{2}A+\frac{1}{2}X$, $\delta(sA, X)$ — deflection of $sA$ and $X$ bodies, $C$ and $\varepsilon_0$ are determined by task $s$, $n$, $r_A$ and $R_A$ ($r_A$ — radius of ball entered in $A$, $R_A$ — described about $A$).