# Upper estimates of Christoffel function on convex domains

## TL;DR

This paper derives new upper bounds for the Christoffel function on convex domains in Euclidean space, using geometric characteristics and algebraic polynomial constructions, with applications demonstrating the bounds' sharpness.

## Contribution

It introduces explicit geometric bounds for Christoffel functions on convex domains and constructs specific polynomials to achieve these bounds, advancing understanding of their behavior.

## Key findings

- New upper bounds for Christoffel function established
- Bounds are expressed in terms of geometric characteristics
- Applications demonstrate the bounds' sharpness

## Abstract

New upper bounds on the pointwise behaviour of Christoffel function on convex domains in ${\mathbb{R}}^d$ are obtained. These estimates are established by explicitly constructing the corresponding "needle"-like algebraic polynomials having small integral norm on the domain, and are stated in terms of few easy-to-measure geometric characteristics of the location of the point of interest in the domain. Sharpness of the results is shown and examples of applications are given.