Abstract:

Let a valuation of a convex polytope $P$ be a linear function with values between 0 and 1 for all points of the polytope. It is a well known fact that any convex polytope $P$ can be described as an intersection of an affine space and a simplex $\Delta$ in high enough dimension. Now consider the question for which polytopes $P$ can every valuation of $P$ be lifted to a valuation of $\Delta$. It was asked by a physicist that also conjectured it to be possible only if $P$ was a simplex itself. The physics interpretation of valuation is models in quantum mechanics, where the simplex gives classical mechanics.

I will present a proof of his conjecture.