A Note on Sharp Multivariate Bernstein- and Markov-Type Inequalities

Let $V$ be a symmetric convex body in ${\mathbb{R}}^m$. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in $V$ and discuss certain properties of the extremal functions. Markov-type inequalities with sharp constants for algebraic polynomials on $V$ and certain non-symmetric convex bodies are proved as well.