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 $n\ge1$ on the unit sphere $\mathbb S^{m-1}$ of the Euclidean space $\mathbb R^m$ for $1\le q<\infty$. We prove that the polynomial $\varrho_n$ 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 $|f|^q$ 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 $t=\xi_m$, $x=(\xi_1,\xi_2,\dots,\xi_m)\in\mathbb S^{m-1}$, is (in a certain sense, unique) extremal in the Nikol'skii inequality on the sphere $\mathbb S^{m-1}$. The corresponding one-dimensional inequalities for algebraic polynomials on a closed interval are discussed.