For , let S be a set of points in in general position. A set I of k points from S is a k-island in S if the convex hull of I satisfies . A k-island in S in convex position is a k-hole in S. For and a convex body of volume 1, let S be a set of n points chosen uniformly and independently at random from K. We show that the expected number of k-holes in S is in . Our estimate improves and generalizes all previous bounds. In particular, we estimate the expected number of empty simplices in S by . This is tight in the plane up to a lower-order term. Our method gives an asymptotically tight upper bound even in the much more general setting, where we estimate the expected number of k-islands in S.

中文翻译：

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随机点集中的空洞和孤岛

对于，令S是处于一般位置的一组点。如果I的凸包满足，则来自S的k个点的集合I是S中的一个k岛。S中凸位置的k岛是S中 的k孔。对于一个体积为 1 的凸体，令S是一组从K中均匀且独立地随机选择 的n个点。我们证明了预期的数量S中的k个孔在. 我们的估计改进并概括了所有先前的界限。特别是，我们通过 估计 S 中空单纯形的预期数量。这在平面上是紧的，直到一个低阶项。即使在更一般的设置中，我们的方法也给出了一个渐近紧密的上界，我们估计了S中k岛的预期数量。