On multivariate polynomials with largest gradients on convex bodies (Q5927571)

Let \(K\subset \mathbb{R}^m\) be a convex body which is symmetric to the origin. The author shows that extremal polynomials with respect to the Markov inequality on \(K\), that is, polynomials \(p_n\) having maximal modulus of the gradient among all polynomials of degree \(\leq n\) with sup-norm equal to 1, coincide with the Chebyshev polynomials along a certain line. More precisely, he shows that for some \(y\) lying on the largest circle around the origin inscribed into \(K\), such an extremal polynomial \(p_n\) satisfies \(p_n(ty)= \pm T_n(t)\) where \(T_n\) is the \(n\)-th Chebyshev polynomial. Moreover, the author applies this result to characterize those \(K\) for which the extremal polynomial is unique (up to a constant multiplier).