Multivariate Splines and Polytopes

TL;DR

This paper introduces a novel approach using multivariate splines to compute polytope volumes, providing explicit formulas and new proofs for existing conjectures, bridging spline theory and polytope geometry.

Contribution

It presents an explicit formula for multivariate truncated power, linking spline theory with polytope volume computation and offering a new proof for Good's conjecture.

Findings

Derived an explicit formula for multivariate truncated power

Connected volume of cube slicing to maximum of box spline

Provided a simple proof for Good's conjecture

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.