Stability of solution of isoperimeter problem in Minkovsky's geometry

V. I. Diskant

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 3 (1996) no. 3, pp. 261-266

Résumé

Let $X$ is a convex body in the $n$-dimensional Minkovsky's space $M^n$ ($n\ge2$) with a symmetrical metric, $B$ – normed body of $M^n$, $I$ – isoperimetrix of $M^n$, $F_B(X)$ – area of the surface, $V_B(X)$ – volume of body $X$ in $M^n$. The theorem was proved: there exist such values of $\varepsilon_0>0$, $C>0$, depending on $n$, $r_I$, $R_I$, that if $F_B^n-n^n V_B(I)V_B^{n-1}(X)<\varepsilon$, $0\le\varepsilon<\varepsilon_0$, $V_B(X)= V_B(I)$ it follow that $\delta_B(X,I)<C\varepsilon^{1/n}$, where $\delta_B(X,I)$ is deviation of $X$ and $I$ in $M^n$, $r_I$ – a capacity coefficient of $B$ in $I$, $R_I$ – scope coefficient of body $I$ by body $B$.