The Heilbronn triangle problem is a classical geometrical problem that asks for a placement of $n$ points in the unit-square $[0,1]^2$, that maximizes the smallest area of a triangle formed by those points. This problem has natural generalizations to higher dimensions. For integers $k, d ge 2$ and a set $mathcal P$ of $n$ points in $[0,1]^d$, let $s = min{(k-1),d}$ and $V_k^{(d)}({mathcal P})$ be the minimum $s$-dimensional volume of the convex hull of $k$ points in $mathcal P$. Here, instead of considering the supremum of $V_k^{(d)}({mathcal P})$, we consider its average value, $vrg{Delta}_k^{(d)}(n)$, when the $n$ points in $mathcal P$ are chosen independently and uniformly at random in $[0,1]^d$. We prove that $vrg{Delta}_k^{(d)}(n) = Theta left(n^{rac{-k}{1+|d-k+1|}}ight)$, for every fixed $k, d ge 2$.
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机译：Heilbronn三角形问题是一个经典的几何问题，要求在Unit-Square $ [0,1] ^ 2 $中的$ n $积分放置，最大化由这些点形成的三角形的最小区域。该问题具有更高尺寸的自然概括。对于Integers $ k，d ge 2 $和set $ mathcal p $ $ n $ points $ [0,1] ^ d $，let $ s = min {（k-1），d $和$ v_k ^ {（d）}（{ mathcal p}）$是$ mathcal p $的凸船部的最低$ s $ -dimimential卷。在这里，而不是考虑$ v_k ^ {（d）}（{ mathcal p}）$，我们考虑其平均值，$ avrg { delta} _k ^ {（d）}（n）$，当$ mathcal p $中的$ n $积分在$ [0,1] ^ d $时独立和均匀地选择。我们证明$ avrg { delta} _k ^ {（d）}（n）= left（n ^ { frac {-k} {1+ | d-k + 1 |}} 右） $，每次固定$ k，d ge 2 $。
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