Chebyshev constant of a compact planar set

All polynomials below are complex formal polynomials with the evaluation and monic conventions of Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials, and $\mathbb{C}$ is identified with ${\mathbb{R}}^2$.

Let $K\subseteq \mathbb{C}$ be nonempty and compact. For a polynomial $p$ put

$\mathrm{\parallel }p{\mathrm{\parallel }}_K:={\sup }_{z\in K}\mathrm{\mid }p(z)\mathrm{\mid }\in [0,\mathrm{\infty })\mathrm{.}$

The supremum is finite and is a maximum: $z\mapsto p(z)$ is entire (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero), hence continuous, and $\mathrm{\mid }\cdot \mathrm{\mid }$ satisfies the modulus laws of Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive; a continuous real-valued function on the nonempty compact space $K$ is bounded and attains its bounds (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value). No choice principle is used: the only selection is that of an extremal point of a continuous function on a compact set, which the cited extreme-value theorem supplies.

For every integer $n\ge 1$ define

$t_n(K):=\inf \{\mathrm{\parallel }p{\mathrm{\parallel }}_K:p\text{ is a monic complex polynomial of degree }n\}\mathrm{.}$

This infimum is a real number: the set is nonempty because $p(Z)=Z^n$ is monic of degree $n$, it consists of nonnegative numbers by the modulus laws, and every nonempty subset of $\mathbb{R}$ bounded below has a greatest lower bound, the infimum (Every nonempty set bounded below has an infimum, Greatest lower bound (infimum)). Thus $0\le t_n(K)<\mathrm{\infty }$ for every $n\ge 1$.

The Chebyshev constant of $K$ is

$\mathrm{cheb}(K):={\inf }_{n\ge 1}{t}_n(K{)}^{1\mathrm{/}n},$

where $t_n(K{)}^{1\mathrm{/}n}$ is the unique nonnegative $n$-th root of $t_n(K)$ (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$). This infimum exists by the same argument: the index set $\{n\in \mathbb{N}:n\ge 1\}$ is nonempty, each $t_n(K{)}^{1\mathrm{/}n}\ge 0$, so the set is nonempty and bounded below by $0$ (Every nonempty set bounded below has an infimum). For the empty set one uses the separate convention

$\mathrm{cheb}(\mathrm{\varnothing }):=0.$

No extremal polynomial is claimed. The quantity $t_n(K)$ is an infimum over polynomials of a fixed degree and is not defined as the norm of a polynomial that attains it. Existence of a best monic polynomial is not asserted, and the definition does not choose one.

Dependence on $K$ only through the sup norms. Every ingredient of the definition is a function of the compact set $K$; for $K=\{a\}$ one has $\mathrm{\parallel }p{\mathrm{\parallel }}_K=\mathrm{\mid }p(a)\mathrm{\mid }$ and $t_n(\{a\})=0$ for all $n\ge 1$ because $(Z-a{)}^n$ is a monic degree-$n$ polynomial vanishing at $a$, so $\mathrm{cheb}(\{a\})=0$.

The root limit is proved later. That the infimum defining $\mathrm{cheb}(K)$ is also the limit of the sequence $t_n(K{)}^{1\mathrm{/}n}$ is the content of The Chebyshev constant is the root limit of monic extremal norms and is not assumed here.