On the monotonicity of the expected volume of a random simplex
On the monotonicity of the expected volume of a random simplex
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关于随机单纯形期望体积的单调性
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发表时间：
2010
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通讯作者：
Luis Rademacher
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作者：
Luis Rademacher
设d维凸体中的随机单形是d+1个随机点的凸船体。研究了随机单形的期望体积作为凸体的函数在包含下是否单调不减的问题。我们证明，如果d是1或2，这成立，如果d >= 4，则不成立。我们还证明了类似的结果，为更高的时刻的随机单形的体积，特别是二阶矩，这对应于凸体的协方差矩阵的行列式。这些问题的动机是切片猜想。
Let a random simplex in a d-dimensional convex body be the convex hull of d+1 random points from the body. We study the following question: As a function of the convex body, is the expected volume of a random simplex monotone non-decreasing under inclusion? We show that this holds if d is 1 or 2, and does not hold if d >= 4. We also prove similar results for higher moments of the volume of a random simplex, in particular for the second moment, which corresponds to the determinant of the covariance matrix of the convex body. These questions are motivated by the slicing conjecture.