Bernstein’s Inequality for Multivariate Polynomials on the Standard Simplex

The classical Bernstein pointwise estimate of the (first) derivative of a univariate algebraic polynomial on an interval has natural extensions to the multivariate setting. However, in several variables the domain of boundedness, even if convex, has a considerable geometric variety. In 1990, Y. Sarantopoulos satisfactorily settled the case of a centrally symmetric convex body by a method we may call “the method of inscribed ellipses.” On the other hand, for the general case of nonsymmetric convex bodies we are only within a constant factor of an exact inequality. The best known results suggest relevance of the generalized Minkowski functional, and a natural conjecture for the exact Bernstein factor was formulated with this geometric quantity. This work deals with the most natural and simple nonsymmetric case, that of a standard simplex in Rd, and computes the exact yield of the method of inscribed ellipses. Although the known general estimates of the Bernstein factor are improved for the simplex here, we find that not even the exact yield of the inscribed ellipse method reaches the conjecture. However, we also show that for an arbitrary convex body the subset of ridge polynomials satisfies the conjecture.