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Bounds on the derivatives of polynomials on centrally symmetric convex bodies. (English. Russian original) Zbl 1102.41011
Izv. Math. 69, No. 3, 607-621 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 3, 179-192 (2005).
The author obtains multidimensional analogues for the Fréchet derivatives of polynomials on convex bodies in a pre-Hilbert space. The starting theorem that plays a key role in this paper gives the inequalities for derivatives of bidimensional polynomials of degree at most \(n\) that map the square in the plane \(\mathbb{R}^2\) to \(\mathbb{R}\). The general multidimensional Markov inequalities are established as consequences of this particular case.

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
46G25 (Spaces of) multilinear mappings, polynomials
52A40 Inequalities and extremum problems involving convexity in convex geometry
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