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.

中文翻译：

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关于随机多胞体的体积阈值的说明

我们研究了通过从给定分布中提取独立同分布点的凸包而生成的随机多胞体的预期体积。我们表明，对于凸体上支持的对数凹面分布，我们需要至少指数数量的（维度上）样本才能使预期体积显着，并且当它们的凹度参数为正时，超指数数量的样本足以用于凹测量。