Tukey Depth Histograms

03/15/2021

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by   Daniel Bertschinger, et al.

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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 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.