Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

We study the sharp Nikol'skii inequality between the uniform norm and L-q norm of algebraic polynomials of a given (total) degree n >= 1 on the unit sphere Sm-1 of the Euclidean space R-m for 1 <= q < infinity. We prove that the polynomial rho(n) in one variable with unit leading coefficient, that deviates least from zero in the space L-q(psi) (-1, 1) of functions f such that vertical bar f vertical bar(q) is summable on (-1, 1) with the Jacobi weight psi(t) - (1 - t)(alpha) (1 + t)(beta), alpha = (m - 1)/2, beta = (m - 3)/2, as a zonal polynomial in one variable t = xi(m), x = (xi(1), xi(2), . . . , xi(m)) is an element of Sm-1, is (in a certain sense, unique) extremal in the Nikol'skii inequality on the sphere Sm-1. The corresponding one-dimensional inequalities for algebraic polynomials on a closed interval are discussed.