Tukey深度直方图
Tukey Depth Histograms
摘要：
平面相对于点集的Tukey深度是一个出现在离散几何和计算几何的许多领域中的概念。特别地，中心点、中心横截、火腿三明治切割或$k$ -边的研究都可以根据相对于一个或多个点集的某些平面的深度来措辞。在这项工作中，我们介绍了Tukey深度直方图$k$ -单位在$\mathbb{R}^d$关于点集$P$，这是一个矢量$D^{k,d}(P)$，其$i$ 'th条目$D^{k,d}_i(P)$表示的$k$ -单位的数量跨越的$k+1$点的$P$具有Tukey深度$i$关于$P$。作为我们的主要结果，我们给出了点的深度直方图的完整特征，即对于任何维度$d$，我们给出了所有可能的直方图$D^{0,d}(P)$的描述。这使我们能够计算出可能的直方图的确切数量。
英文摘要：
The Tukey depth of a flat with respect to a point set is a concept that appears in many areas of discrete and computational geometry. In particular, the study of centerpoints, center transversals, Ham Sandwich cuts, or $k$-edges can all be phrased in terms of depths of certain flats with respect to one or more point sets. In this work, we introduce the Tukey depth histogram of $k$-flats in $\mathbb{R}^d$ with respect to a point set $P$, which is a vector $D^{k,d}(P)$, whose $i$'th entry $D^{k,d}_i(P)$ denotes the number of $k$-flats spanned by $k+1$ points of $P$ that have Tukey depth $i$ with respect to $P$. As our main result, we give a complete characterization of the depth histograms of points, that is, for any dimension $d$ we give a description of all possible histograms $D^{0,d}(P)$. This then allows us to compute the exact number of possible such histograms.
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