The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -along with its equality conditions. 2010 Mathematics Subject Classification. 52A40, 52B11.
Ehrhart's famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors. It turns out that the number of integral points can still be written as a rational quasi-polynomial. Furthermore, the coefficients of this rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation.
Given a matrix A ∈ Z m×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Z m of all vectors b ∈ Z m such that the associated knapsack polytope: Ax = b} contains at least s integer points. In this paper we investigate the structure of the set Fs(A) using the concept of s-covering radius. In particular, in the case m = 1 we prove an optimal lower bound for the s-Frobenius number, which is the largest integer b such that P (A, b) contains less than s integer points.2010 Mathematics Subject Classification. Primary: 90C10, 52C07, 11D07; Secondary: 90C27, 11H06.
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