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Colored Tverberg Applet(a.k.a. 'in search for a counter-example') | ||||
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Theorem [Tverberg's theorem]:
Given positive integers d and p, any (p-1)(d+1)+1 point in Rd can be partitioned into p parts whose convex hulls intersect.
Theorem [Colored Tverberg theorem, BMZ version]: Suppose furthermore that the points are colored in such a way that every color appears at most p-1 times. Under the condition that p is a prime, the parts in Tverberg's theorem can be chosen to be rainbow colored (i.e. every part uses every color at most once). It is natural to conjecture that this holds for all p>0, not only for primes. You can use the above applet to search for counter-examples for p=4. (Let me know if you find one!) Sierksma conjectured that the number of Tverberg partitions is always at least (p-1)!d, it is also called the Dutch Cheese problem since he offered a Dutch cheese for the solution. His bound is obtained for example by the standard point configuration: Put p-1 points on each vertex of a d-simplex, and one further point in the center. It seems reasonable to conjecture a similar lower bound for the colored Tverberg theorem, but note that the number of colored partitions depends on how the colors are distributed. Links:
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