# Sharp Constants of Approximation Theory. III. Certain Polynomial Inequalities of Different Metrics on Convex Sets

## TL;DR

This paper establishes limit relations between sharp polynomial inequalities on convex bodies and entire functions of exponential type, advancing the understanding of approximation theory in multivariate settings.

## Contribution

It introduces new limit relations connecting polynomial inequalities on convex bodies with entire functions of exponential type, extending previous univariate results.

## Key findings

- Limit relations between polynomial and entire function inequalities
- Sharp constants characterized for multivariate convex bodies
- Advancement in approximation theory for multivariate polynomials

## Abstract

Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on $V^*$ and the corresponding constants for entire functions of exponential type with the spectrum in $V$.