Practical Polytope Volume Approximation

Polytope approximation and the Mahler volume

The problem of approximating convex bodies by polytopes is an important and well studied problem. Given a convex body K in R, the objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error ε. Results to date have been of two types. The first type assumes that K is smooth, and bounds hold in the limit as ε tends ...

On Padé Approximation of Some Practical Functions

On the approximation of a polytope

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P ∈ R by its dual Lp-centroid bodies is independent of the geometry of P . In particular we show that if P has volume 1, lim p→∞ p log p „ |Z◦ p (P )| |P ◦| − 1 « = n. We provide an application to the approximation of polytopes by uniformly convex sets.

On the Volume of a Certain Polytope

Let n ≥ 2 be an integer and consider the set Tn of n× n permutation matrices π for which πij = 0 for j ≥ i+ 2. In this paper we study the convex hull of Tn, which we denote by Pn. Pn is a polytope of dimension ( n 2 ) . Our main purpose is to provide evidence for the following conjecture concerning its volume. Let vn denote the minimum volume of a simplex with vertices in the affine lattice spa...

The asymptotic volume of the Birkhoff polytope

Let m,n ≥ 1 be integers. Define Tm,n to be the transportation polytope consisting of the m × n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case Bn = Tn,n is the much-studied Birkhoffvon Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determ...