On Multivariate Polynomials with Largest Gradients on Convex Bodies

In this paper we study the extremal polynomials for the Markov inequality on a convex symmetric body K m, that is, norm 1 polynomials on K whose gradients attain the largest value on K. It is shown that any such polynomial must coincide with the Chebyshev polynomial along a certain line. Moreover, this fact is applied to the study of uniqueness of extremal polynomials