Mathematics Colloquia and Seminars

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The colorful Caratheodory theorem states that for any $d+1$ sets of points in $d$-space whose convex hulls
contain the origin, there is a transversal whose convex hull also contains the origin. How many such transversals
are there? This number, called the colorful simplicial depth, was introduced and studied by Deza et al (2006). In
particular, they conjectured that there are at most $1 + d^{d+1}$ such transversals whenever each set contains
$d+1$ points.
Consider the Minkowski sum of $d-1$ general triangles in $d$-space. Some codimension-$1$
faces (facets) of it will be parallelpipeds. How many are there? In the context of normal surface theory Burton (2003)
conjecture that there are at most $1 + 2^{d-1}$ such facets.
Both bounds look similar. In this talk I want to explain the connection (using generalized Gale transforms)
and sketch the proof of a more general upper bound theorem (using combinatorial topology). This is joint work with
Adiprasito, Brinkmann, Padrol, Paták, and Patáková.