Analogues of the Markov and Bernstein inequalities on convex bodies in Banach spaces

We obtain analogues of the A. A. Markov and Bernstein inequalities for algebraic polynomials on convex bounded closed bodies in Banach spaces. For centrally symmetric bodies we establish an exact estimate in Markov's inequality, an estimate in Bernstein's inequality that is asymptotically exact for polynomials on the ball, and estimates for the norms of higher derivatives.