Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies
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- Révész, S. Constr. Approx. (2001) 17: 465. doi:10.1007/s003650010043
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For a compact set K\subset Rd with nonempty interior, the Markov constants Mn(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 Mn(K) = n2/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 Bd(\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1]d the extremal polynomials are essentially unique.