We prove that there exists an absolute constant $\alpha >1$ with thefollowing property: if $K$ is a convex body in ${\mathbb R}^n$ whose center ofmass is at the origin, then a random subset $X\subset K$ of cardinality ${\rmcard}(X)=\lceil\alpha n\rceil $ satisfies with probability greater than$1-e^{-n}$ {K\subseteq c_1n\,{\mathrm conv}(X),} where $c_1>0$ is an absoluteconstant. As an application we show that the vertex index of any convex body$K$ in ${\mathbb R}^n$ is bounded by $c_2n^2$, where $c_2>0$ is an absoluteconstant, thus extending an estimate of Bezdek and Litvak for the symmetriccase.
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机译：我们证明存在具有以下属性的绝对常数$ \ alpha> 1 $：如果$ K $是$ {\ mathbb R} ^ n $中的凸体，其质心在原点，则有一个随机子集$ X \ subset K $基数$ {\ rmcard}（X）= \ lceil \ alpha n \ rceil $满足概率大于$ 1-e ^ {-n} $ {K \ subseteq c_1n \，{\ mathrm conv}（ X），}，其中$ c_1> 0 $是绝对常数。作为一个应用程序，我们证明$ {\ mathbb R} ^ n $中任何凸体$ K $的顶点索引都由$ c_2n ^ 2 $界定，其中$ c_2> 0 $是绝对常数，因此扩展了Bezdek和Litvak用于对称情况。
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