Bernstein Type Theorems for Compact Sets in Rn Revisited Original Research Article

In this paper we complete some results of (J. Approx. Theory69 (1992), 156-166) and give a geometrical approach to the multivariate Bernstein and Markov inequalities. The most interesting and slightly surprising result is a sharp Markov inequality for convex symmetric subsets of Rn formulated in geometrical language. A sharp inequality for gradients of polynomials extends an old Kellog result (Math. Z.27 (1927), 55-64), and it is also a partial positive answer to a question formulated by Wilhelmsen (J. Approx. Theory11 (1974), 216-220) in 1974.