CONVEXITY AND BERNSTEIN POLYNOMIALS ON k-SIMPLOIDS

This paper is concerned with Bernstein polynomials on k-simploids by which we mean a crossproduct of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials ofa given function f on a k-simploid form a decreasing sequence then f+l, where l is any correspondingtensor product of affine functions. achieves its maximum on the boundary of the k-simploid. Thisextends recent results in 1] for bivariate Bernstein polynomials on triangles. Unlike the approachused in 1] our approach is based on semigroup techniques and the maximum principle for secondorder elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.