Mathematics Colloquia and Seminars

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The simplicial depth of a point p in R^d with respect to a finite set S of points is the number of (d+1)-sets from S whose convex hull contains p. A natural generalization is to colour the points of S and consider only the colourful simplices containing p. We exhibit a configuration where p is in the convex hull of each of (d+1) colours, but is only in d^2+1 colourful simplices. We conjecture that this is minimal and investigate lower bounds. This result can be viewed as a refinement of Barany's Colourful Caratheodory Theorem.

This talk was scheduled in place of Andrew Critch's talk which was canceled.