Abstract
We prove that there exists an absolute constant \({\alpha > 1}\) with the following 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 \({{\rm card}(X)=\lceil\alpha n\rceil }\) satisfies with probability greater than \({1-e^{-c_1n}}\)
where \({c_1, c_2 > 0}\) are absolute constants. As an application we show that the vertex index of any convex body K in \({{\mathbb R}^n}\) is bounded by \({c_3n^2}\), where \({c_3 > 0}\) is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.
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Brazitikos, S., Chasapis, G. & Hioni, L. Random approximation and the vertex index of convex bodies. Arch. Math. 108, 209–221 (2017). https://doi.org/10.1007/s00013-016-0975-2
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DOI: https://doi.org/10.1007/s00013-016-0975-2