Volumes of polytopes via power series

Hopefully we know that i0xi =11x. Similarly one computes that i1xi =x2x1. Interestingly, 11x+x2x1=1+x which is the sum corresponding to the integers in the interval [0,1]=[0,][,1]. We will explain generalization of this (called Brion's theorem) to integer points in convex polytopes of arbitrary dimension. Surprisingly, this gives a formula for the volume of a polytope in terms of summing up certain rational functions associated to vertices of the polytope. We also discuss related theorems of Lawrence-Varchenko and Brianchon-Gram about characteristic function of a convex polytope.

Tuesday, April 9, 2019 - 12:00 to 13:00

Thackeray 703

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Kiumars Kaveh

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