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  "body": "\n## Statement\n\nLet $n \\ge 0$ be an integer and let $\\sigma \u003e 0$.\nLet $p$ be a real polynomial of [[degree|0DPF6WRZW6]] at most $n$.\nLet $E \\subseteq [-1, 1 + \\sigma]$ be a Lebesgue measurable set of measure at least $2$ such that $\\abs{p(x)} \\le 1$ for all $x \\in E$.\nThen\n$$\\max_{-1 \\le x \\le 1 + \\sigma} \\abs{p(x)} \\le T_n(1 + \\sigma),$$\nwhere $T_n$ is the [[Chebyshev polynomial|M6EAR1JHPP]] of the first kind of degree $n$.\n\n## Remarks\n\nThe bound is sharp: $p = T_n$ and $E = [-1, 1]$ give equality, since $T_n$ is increasing on $[1, \\infty)$.\nProof idea: compare $p$ with a rescaled Chebyshev polynomial whose set where it is at most $1$ in absolute value has measure $2$.\nA count of sign changes, as in the minimal deviation property of $T_n$, shows that $p$ cannot be larger.\n\nExample: for $n = 1$, a line with $\\abs{p} \\le 1$ on a set of measure $2$ in $[-1, 1 + \\sigma]$ is at most $1 + \\sigma = T_1(1 + \\sigma)$ in absolute value on the whole interval.\n\nBy an affine change of variable, for an interval $I$ and a measurable $E \\subseteq I$ of positive measure,\n$$\\sup_{I} \\abs{p} \\le T_n\\left(\\frac{2 \\abs{I}}{\\abs{E}} - 1\\right) \\sup_{E} \\abs{p} \\le \\left(\\frac{4 \\abs{I}}{\\abs{E}}\\right)^n \\sup_{E} \\abs{p}.$$\nWithout a lower bound on the measure of $E$ there is no bound: $p(x) = c x$ is small on a tiny interval around $0$ for every $c$.\n\n## Explanations\n\n### L1 human\n\nA polynomial of degree $n$ that is small on a large part of an interval cannot be huge on the rest of it.\nThe Remez inequality gives the exact limit, and the Chebyshev polynomial is the worst case.\n\n### L2 human\n\nThe inequality is a quantitative form of the fact that a polynomial is determined by its values on any set of positive measure.\nIt extends Chebyshev's extremal property, which concerns the growth of $T_n$ outside $[-1, 1]$, to arbitrary measurable sets.\nIt is used to bound the size of sublevel sets of polynomials and has versions for several variables and for exponential sums.\n",
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