Analogues of the Markov and Bernstein inequalities for polynomials in Banach spaces

We propose analogues of the inequalities of A. A. and V. A. Markov for algebraic polynomials on the cube in $\mathbb R^m$ in the $l_2^{(m)}$ and $l_1^{(m)}$ metrics, as well as for polynomials on closed bounded convex bodies in a Banach space and on centrally symmetric bodies of the same type. For the last two cases analogues of Bernstein's inequality are obtained.