Sharp Constants of Approximation Theory. V. An Asymptotic Equality Related to Polynomials with Given Newton Polyhedra

Let $V\subset {\mathbb{R}}^m$ be a convex body, symmetric about all coordinate hyperplanes, and let ${\text{PP}}_{aV},a\ge 0$, be a set of all algebraic polynomials whose Newton polyhedra are subsets of $aV$. We prove a limit equality as $a\to \text{iy}$ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from ${\text{PP}}_{aV}$ and the corresponding constant for entire functions of exponential type with the spectrum in $V$.