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Stability of Minkowski and Brunn's equations solutions

V. I. Diskant

Cherkasy Institute of Industrial Engineering

Abstract: 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 $A$).

Received: 07.04.1997

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