Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies
- Szilárd Révész
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For a compact set K\subset R d with nonempty interior, the Markov constants M n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K .
It is known that for convex, symmetric bodies M n (K) = n 2 /r(K) , where r(K) is the ``half-width'' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] d the extremal polynomials are essentially unique.
- Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies
Volume 17, Issue 3 , pp 465-478
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- Key words. Markov inequality, Multivariate polynomials, Symmetric convex bodies, Supporting hyperplanes. AMS Classification. 41A17, 41A63, 41A10.
- Szilárd Révész (A1)
- Author Affiliations
- A1. Alfréd Rényi Mathematical Institute of the Hungarian Academy of Sciences Budapest Reáltanoda u. 13—15 1053 Hungary email@example.com, HU