Advanced Optimization Laboratory Title : Colourful Simplicial
نویسندگان
چکیده
Inspired by Bárány’s colourful Carathéodory theorem [Bár82], we introduce a colourful generalization of Liu’s simplicial depth [Liu90]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any ddimensional configuration is d + 1 and that the maximum is d + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.
منابع مشابه
Advanced Optimization Laboratory Title : Computational Lower Bounds for Colourful Simplicial
The colourful simplicial depth problem in dimension d is to find a configuration of (d +1) sets of (d +1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d 2 +1 simplices is known, and is conjectured to be minimal. This has been confirme...
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Inspired by Bárány’s Colourful Carathéodory Theorem [4], we introduce a colourful generalization of Liu’s simplicial depth [13]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 +1 and that the maximum is dd+1 +1. We exhibit configurations attaining each of these depths, and apply our results to the ...
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