Bernstein's inequality for multivariate polynomials on the standard simplex.

Journal of Inequalities and Applications | January 1, 2005| | Copyright

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 ...

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