Journal of Inequalities and Applications
Volume 2005 (2005), Issue 2, Pages 145-163
doi:10.1155/JIA.2005.145
Bernstein's inequality for multivariate polynomials on the standard simplex
1Department of Mathematics, University of Sofia, Boulevard James Boucher 5, Sofia 1164, Bulgaria
2A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127 1364, Budapest, Hungary
Received 31 July 2003
Copyright © 2005 Lozko B. Milev and Szilárd Gy. Révész. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.