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  "body": "\n## Definition\n\nLet $K$ be a [[compact set|0R6EWFKRCF]] in $\\C$ with at least one point.\nFor a polynomial $p$, let $\\norm{p}_K = \\max_{z \\in K} \\abs{p(z)}$.\nFor an integer $n \\ge 1$, let\n$$t_n(K) = \\inf \\set{ \\norm{p}_K : p \\text{ a monic polynomial of degree } n }.$$\nThe *Chebyshev constant* of $K$ is $\\tau(K) = \\lim_{n \\to \\infty} t_n(K)^{1/n}$.\n\n## Remarks\n\nThe infimum is attained, and a [[monic polynomial|0C3QBCR46M]] $p$ of degree $n$ with $\\norm{p}_K = t_n(K)$ is a Chebyshev polynomial of $K$.\nSince a product of monic polynomials is monic, $t_{m+n}(K) \\le t_m(K) \\, t_n(K)$.\nSo the sequence $\\log t_n(K)$ is subadditive, and [[Fekete's lemma|8A4GGX1MEY]] shows that the limit exists and equals $\\inf_n t_n(K)^{1/n}$.\n\nExample: let $K$ be the unit circle.\nFor a monic $p(z) = z^n + a_{n-1} z^{n-1} + \\cdots + a_0$, Parseval's identity gives $\\frac{1}{2\\pi} \\int_0^{2\\pi} \\abs{p(e^{i\\theta})}^2 \\, d\\theta = 1 + \\sum_k \\abs{a_k}^2 \\ge 1$.\nSo $\\norm{p}_K \\ge 1$, with equality for $p = z^n$, and $t_n(K) = 1$ and $\\tau(K) = 1$.\n\nNon-example: the polynomial $2 z^n$ has maximum $2$ on the unit circle, but it is not monic, so it does not enter the infimum.\nWithout the monic condition the infimum would be $0$.\n\nIf $K$ is a finite set with $m$ points, then $p(z) = \\prod_{a \\in K} (z - a)$ vanishes on $K$, so $t_n(K) = 0$ for $n \\ge m$ and $\\tau(K) = 0$.\n",
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