# Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies

## Authors

- First Online:

DOI: 10.1007/s003650010043

- Cite this article as:
- Révész, S. Constr. Approx. (2001) 17: 465. doi:10.1007/s003650010043

- 20 Views

## Abstract.

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.