论文标题
海尔布隆在更高维度中的三角形问题的上限
Upper bounds for Heilbronn's triangle problem in higher dimensions
论文作者
论文摘要
我们开发了一种新的简单方法，以证明在更高维度中Heilbronn的三角问题的概括。除其他事项外，我们显示以下内容：对于固定$ d \ ge 1 $，$ [0，1]^d $ size $ n $包含的任何子集 - $ d+ 1 $点，最多跨越音量的简单$ c_d n^{ - \ log d+ 6} $， - $ 1.1 d $点的凸赫尔最多具有$ c_d n^{ - 1.1} $， - $ k \ ge 4 \ sqrt {d} $ points跨越$（k-1）$ - 尺寸简单量最多最多$ c_d n^{ - \ frac {k-1} {d} {d} {d} - \ frac {k^2} {8d^2}}} $。
We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$ contains - $d+1$ points which span a simplex of volume at most $C_d n^{-\log d+ 6}$, - $1.1 d$ points whose convex hull has volume at most $C_d n^{-1.1}$, - $k\ge 4\sqrt{d}$ points which span a $(k-1)$-dimensional simplex of volume at most $C_d n^{-\frac{k-1}{d} - \frac{k^2}{8d^2}}$.
