A note on volume thresholds for random polytopes

Abstract

We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at least exponentially many (in dimension) samples for the expected volume to be significant and that super-exponentially many samples suffice for concave measures when their parameter of concavity is positive.

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