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  "body": "\n## Statement\n\nLet $\\Omega \\subset \\R^n$ be a bounded open set of diameter $d$.\nAssume $\\Omega$ is star-shaped with respect to every point of an open ball $B \\subset \\Omega$ of radius $\\rho \u003e 0$.\nLet $m \\ge 1$ be an integer and $1 \\le p \\le \\infty$.\nLet $u$ be in the [[Sobolev space|M4CEZ5K9HP]] $W^{m,p}(\\Omega)$ and let $0 \\le k \\le m$.\nLet $\\abs{u}_{W^{k,p}(\\Omega)}$ be the [[seminorm|4KX81Q85DT]] formed from the $L^p$ norms of the [[weak derivatives|C3EC148VZD]] of $u$ of order exactly $k$.\nThen there is a constant $C$, depending only on $n$, $m$, $p$ and $d / \\rho$, with the following property.\nFor every $u \\in W^{m,p}(\\Omega)$ there is a [[polynomial|07GNX8GCYP]] $q$ of degree less than $m$ such that\n$$\\abs{u - q}_{W^{k,p}(\\Omega)} \\le C \\, d^{\\,m - k} \\, \\abs{u}_{W^{m,p}(\\Omega)} \\quad \\text{for } k = 0, 1, \\ldots, m.$$\n\n## Remarks\n\nOne can take for $q$ the averaged Taylor polynomial of $u$ of order $m$.\nIt is the Taylor polynomial of degree $m - 1$ about a point $y$, averaged over $y \\in B$ with a smooth weight.\nThe remainder of this averaged Taylor formula is an integral operator applied to the derivatives of order $m$ of $u$, and its bounds give the estimate.\nThe scaling $d^{\\,m-k}$ follows by mapping $\\Omega$ to a set of diameter $1$.\n\nThe right side involves only the seminorm of order $m$, with no lower-order terms.\nEvery polynomial of degree less than $m$ has $\\abs{q}_{W^{m,p}} = 0$, so such a bound can only control $u$ up to these polynomials, which is why $q$ appears.\n\nExample: for $m = 1$ the lemma gives a constant $c$ with $\\norm{u - c}_{L^p(\\Omega)} \\le C d \\, \\norm{\\nabla u}_{L^p(\\Omega)}$, a Poincaré-type inequality.\n\nIn finite element analysis, the lemma is applied on a reference element to the error $u - I u$ of an interpolation operator $I$ that reproduces polynomials of degree less than $m$.\nSince $u - I u = (u - q) - I(u - q)$, the error is bounded by the seminorm $\\abs{u}_{W^{m,p}}$, and scaling back to an element of size $h$ gives the factor $h^{m-k}$.\n\n## Explanations\n\n### L1 human\n\nA function whose derivatives of order $m$ are small is close to a polynomial of degree less than $m$.\nThe lemma makes this precise in Sobolev norms, with a constant that only depends on the shape of the domain.\nIt is the basic tool behind the error estimates of the finite element method.\n\n### L2 human\n\nFor a triangle of size $h$ and piecewise linear interpolation, the lemma with $m = 2$ gives errors of order $h^2$ in $L^2$ and of order $h$ in $H^1$, for $u \\in H^2$.\nThe constant depends on $d / \\rho$, which is why finite element meshes must avoid very thin elements.\nCombined with Céa's lemma, it turns approximation by polynomials into convergence rates of the finite element method.\n",
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