Extremal functions for real convex bodies

TL;DR

This paper investigates the smoothness of extremal functions for convex bodies in real space and relates their equilibrium measures to complex analysis tools, using extremal ellipses as a key method.

Contribution

It introduces new results on the smoothness of extremal functions and establishes a formula linking equilibrium measures of convex bodies to their Robin indicatrix.

Findings

Smoothness properties of extremal functions in $ ext{R}^2$

A formula connecting equilibrium measures in $ ext{R}^n$ and Robin indicatrix

Use of extremal ellipses as a main analytical tool

Abstract

We study the smoothness of the Siciak-Zaharjuta extremal function associated to a convex body in $\mathbb{R}^2$. We also prove a formula relating the complex equilibrium measure of a convex body in $\mathbb{R}^n$ to that of its Robin indicatrix. The main tool we use are extremal ellipses.