Multivariate Bernstein polynomials and convexity

It is well known that in two or more variables Bernstein polynomi-als do not preserve convexity. Here we introduce two variations, one stronger than the classical notion, the other one weaker, which are preserved. Moreover, a weaker suucient condition for the monotony of subsequent Bernstein polynomials is given. linearly independent, in the course of which d has to be greater than or equal to m. A point p in the aane hull of p 0 ; : : : ; p m can be uniquely written as p = m X k=0 u k p k ; where u 0 + + u m = 1: The coeecients of u = (u 0 ; : : :; u m) 2 R m+1 are called the barycentric coordinates of p with respect to p 0 by using the 1{1 mapping which associates each point of the simplex with its uniquely deened barycentric coordinates. We consider S m = f(x 1 ; : : : ; x m) : x k 0; x 1 + + x m 1g to be the standard unit simplex in R m. If f is a function deened on an arbitrary m{ dimensional simplex, especially on S m , it will prove quite useful to consider f as a function deened over S m by the use of barycentric coordinates.