University of Birmingham > Talks@bham > Optimisation and Numerical Analysis Seminars > Volumes of alcoved polyhedra and Mahler's conjecture

Volumes of alcoved polyhedra and Mahler's conjecture

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If you have a question about this talk, please contact Sergey Sergeev.

There is a conjecture, by K. Mahler, open since 1939 in dimensions 3 and higher: every centrally symmetric convex body K satisfies the inequality “vol(K) vol(K’) >= 4^n/n!” where K’ is the polar of K. The conjecture means that the volume product vol(K) vol(K’) is minimal when K is an n-dimensional cube (or an affine transform of it). On the other hand, the Blaschke÷≠-Santalo inequality (proved in 1949) says that the volume product is maximal when K is an n-dimensional sphere (or an ellipsoid, an affine transform).

A polytope is the generalization to n-dimensional space of a polygon (n=2) and a polyhedron (n=3). Alcoved polytopes are polytopes having facets of only two types: x_i=cnt and x_i-x_j=cnt. Each n-dimensional alcoved polytope P can be represented by a square matrix A(P) of order n+1. The matrix A(P) can be assumed to be multiplicatively idempotent with respect to tropical product, where “max” plays the role of addition (tropical sum) and “plus” plays the role of multiplication (tropical product). The volume vol(P) is obtained as an expression in terms of the entries A(P). In the case n=3, we have found a general formula for such expression. Further, for centrally symmetric P, we have found another formula for the volume product vol(P)vol(P’) (after an affine change). This is a rational expression in three variables (with rational coefficients). Mahler’s Conjeture in this case reduces to the statement that a certain polynomial, which is non-homogeneous, of degree 6, in three variables, is non—negative over a given tetrahedron. We will show that this holds true.

In October 2017, a second version of a proof of Mahler Conjecture has been uploaded to Arxiv, for n=3, by Iriyeh and Shibata (a first version in June 2017).

This talk is part of the Optimisation and Numerical Analysis Seminars series.

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