Random approximation and the vertex index of convex bodies (Q515532)

The authors prove (Theorem 1.2) that there exists an absolute constant \(\alpha > 1\) with the property: if \(K\) is a convex body in \(\mathbb{R}^n\) whose center of mass is at the origin, then a random subset \(X \subset K\) of cardinality \(\mathrm{card}(X) = \lceil \alpha n \rceil\) satisfies with probability greater than \(1- e^{-c_1 n}\) \[ K \subseteq c_2 n \;\mathrm{conv}(X), \] where \(c_1, c_2 > 0\) are absolute constants. Building on this result, the authors extend the definition of the vertex index of a symmetric convex body to the case of not necessarily symmetric \(n\)-dimensional convex bodies and show (Theorem 1.3) that this index is upper bounded by \(c_3 n^2\), where \(c_3\) is an absolute constant. This result extends an estimate of \textit{K. Bezdek} and \textit{A. E. Litvak} for the symmetric case [Adv. Math. 215, No. 2, 626--641 (2007; Zbl 1131.46011)].