On the existence of regions with minimal third degree integration formulas

Abstract:

A. H. Stroud has shown that $n + 1$ is the minimum possible number of nodes in an integration formula of degree three for any region in ${E_n}$. In this paper, in answer to the question of the attainability of this minimal number, we exhibit for each $n$ a region that possesses a third degree formula with $n + 1$ nodes. This is accomplished by first deriving an $(n + 2)$-point formula of degree three for an arbitrary region that is invariant under the group of affine transformations that leave an $n$-simplex fixed. The formula is then applied to a one-parameter family of such regions, and a value of the parameter is determined for which the weight at the centroid vanishes.