  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.
