Least multivariate Chebyshev polynomials on diagonally determined sets

Abstract

We consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval $[-1,1]$. Let $\Pi^*_n$ be the subset of polynomials of degree at most $n$ in $d$ variables, whose homogeneous part of degree $n$ has coefficients summing up to $1$. The problem is determining a polynomial in $\Pi^*_n$ with the smallest uniform norm on a domain $\Omega$, which we call a least Chebyshev polynomial (associated with $\Omega$). Our main result solves the problem for $\Omega$ belonging to a non-trivial class of sets that we call diagonally-determined, and establishes the remarkable result that a least Chebyshev polynomial can be given via the classical, univariate, Chebyshev polynomial. In particular, the solution can be independent of the dimension. Diagonally-determined domains include centered balls in $\mathbb{R}^d$ in any norm, but can be non-convex and even non-simply connected. We also introduce a computational procedure, based on semidefinite programming hierarchies, to detect if a given semi-algebraic set is diagonally-determined.