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Bernstein’s inequality for multivariate polynomials on the standard simplex. (English) Zbl 1082.41010

If a univariate algebraic polynomial \(p\) of a certain degree is given, then the classical Bernstein-Szegö inequality provides a sharp upper bound for the derivative of \(p\) in functions of \(p\) itself. For multivariate polynomials the exact inequality (i.e. the upper bound is sharp) is known only for symmetric convex bodies. This result is due to Y. Sarantopoulos [Math. Proc. Camb. Philos. Soc. 110, No. 2, 307–312 (1991; Zbl 0761.46035)]. Furthermore a natural conjecture exists for the exact Bernstein inequality which relates the sharp upper bound to the generalized Minkowski functional [see Sz. Gy. Révész and Y. Sarantopoulos, J. Convex Anal. 11, No. 2, 303–334 (2004; Zbl 1068.46008)].
In this paper the autors investigate the Bernstein inequality for multivariate polynomials on the most natural and simple nonsymmetric convex body: the standard simplex. The known general estimates of the Bernstein inequality for the simplex are improved here, but the conjecture is not reached, which suggests that their bound is not sharp. The authors also show that for an arbitary convex body the conjecture is always satisfied when using ridge polynomials.
Reviewer: Jan Maes (Leuven)

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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