Multivariate Splines and Polytopes
Numerical Analysis
2010-10-19 v4 Combinatorics
Metric Geometry
Abstract
In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion's formula for the volume of polytopes. We also prove that the integration of polynomials over polytopes can be dealt with by the multivariate truncated power. Moreover, we show that the volume of the cube slicing can be considered as the maximum value of the box spline. Based on this connection, we give a simple proof for Good's conjecture, which has been settled by probability methods.
Cite
@article{arxiv.0806.1127,
title = {Multivariate Splines and Polytopes},
author = {Zhiqiang Xu},
journal= {arXiv preprint arXiv:0806.1127},
year = {2010}
}