Stability of the Minkowski Measure of Asymmetry for Convex Bodies

Given a convex body $C\subset R^n$ (i.e., a compact convex set with nonempty interior), for $x\in$ {\it int}$(C)$, the interior, and a hyperplane $H$ with $x\in H$, let $H_1,H_2$ be the two support hyperplanes of $C$ parallel to $H$. Let $r(H, x)$ be the ratio, not less than 1, in which $H$ divides the distance between $H_1,H_2$. Then the quantity $${\it As}(C):=\inf_{x\in {\it int}(C)}\,\sup_{H\ni x}\,r(H,x)$$ is called the Minkowski measure of asymmetry of $C$. {\it As}$(\cdot)$ can be viewed as a real-valued function defined on the family of all convex bodies in $R^n$. It has been known for a long time that {\it As}$(\cdot)$ attains its minimum value 1 at all centrally symmetric convex bodies and maximum value $n$ at all simplexes. In this paper we discuss the stability of the Minkowski measure of asymmetry for convex bodies. We give an estimate for the deviation of a convex body from a simplex if the corresponding Minkowski measure of asymmetry is close to its maximum value. More precisely, the following result is obtained: Let $C\subset R^n$ be a convex body. If {\it As}$(C)\ge n-\varepsilon$ for some $0\le \varepsilon < 1/8(n+1),$ then there exists a simplex $S_0$ formed by $n+1$ support hyperplanes of $C$, such that $$(1+8(n+1)\varepsilon)^{-1}S_0\subset C\subset S_0,$$ where the homethety center is the (unique) Minkowski critical point of $C$. So $$d_{{\rm BM}}(C,S)\le 1+8(n+1)\varepsilon$$ holds for all simplexes $S$, where $d_{{\rm BM}}(\cdot,\cdot)$ denotes the Banach-Mazur distance.