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Colorful simplicial depth

by · Jul 23, 2016 · 183 views ·

Given d+1 sets S1,S2,…,Sd+1 (called color classes) in Rd, a simplex is called colorful, if all its vertices are in different color classes. The number of colorful simplices containing a point p∈Rd is known as the colorful simplicial depth of p. Since the point with the maximal depth can be seen as a higher dimensional analogue to median, the colorful simplicial depth is studied not only in discrete and computational geometry, but also in statistics and data analysis. The first result concerning colorful simplicial depth in discrete geometry was the colorful Carathéodory's theorem by Bárány in 1982: ``Any point p∈Rd contained in the convex hull of all color classes has a non-zero simplicial depth provided that each color class has at least d+1points.'' In 2006 Deza et al. asked for the minimal and maximal values of the colorful simplicial depth of the point p in colorful Carathéodory's theorem. We use methods from combinatorial topology to prove a tight upper bound of the form 1+∏d+1i=1(|Si|−1). Joint work with Karim Adiprasito, Philip Brinkmann, Arnau Padrol, Pavel Paták, and Raman Sanyal.

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