Uniqueness of Markov-extremal polynomials on symmetric convex bodies

For a compact set K subset of R-d with nonempty interior, the Markov consta nts M-n(K) can be defined as the maximal possible absolute value attained o n 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), whe re 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, a nd show that they are essentially unique if and only if K has a certain geo metric property, called flatness. For example, for the unit ball B-d (0,1) If we do not have uniqueness, while for the unit cube [-1, 1](d) the extrem al polynomials are essentially unique.