Convexity and Bernstein polynomials on k-simploids

This paper is concerned with Bernstein polynomials on k-simploids by which we mean a cross product of k lower dimensional simplices. Specifically, we show that if the Bernstein polynomials of a given function f on a k-simploid form a decreasing sequence then f +l, where l is any corresponding tensor product of affine functions, achieves its maximum on the boundary of the k-simploid. This extends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approach used in [1] our approach is based on semigroup techniques and the maximum principle for second order elliptic operators. Furthermore, we derive analogous results for cube spline surfaces. © 1990 Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A.