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  "body": "\n## Statement\n\nLet $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension $n \\ge 2$.\nConsider the compact smooth hypersurfaces $\\Sigma$ that divide $M$ into two disjoint open sets $A$ and $B$ with $M = A \\cup \\Sigma \\cup B$.\nLet $\\operatorname{area}(\\Sigma)$ be the $(n-1)$-dimensional volume of $\\Sigma$.\nThe *Cheeger constant* of $M$ is\n$$h(M) = \\inf_{\\Sigma} \\frac{\\operatorname{area}(\\Sigma)}{\\min(\\operatorname{vol}(A), \\operatorname{vol}(B))}.$$\nLet $\\lambda_1$ be the smallest nonzero eigenvalue of the [[Laplace–Beltrami operator|F4NWZNTRMD]] of $M$.\nThen\n$$\\lambda_1 \\ge \\frac{h(M)^2}{4}.$$\n\n## Remarks\n\nThe proof starts from the variational characterisation of $\\lambda_1$ as the infimum of $\\int_M \\abs{\\nabla f}^2 / \\int_M f^2$ over nonzero smooth functions $f$ with $\\int_M f = 0$.\nOne applies the coarea formula to $g^2$, where $g$ is the positive part of an eigenfunction shifted so that its support has at most half the volume.\nThe bound from $h(M)$ on each level set, together with the Cauchy–Schwarz inequality, gives the factor $1/4$.\nThe inequality shows that a small $\\lambda_1$ forces a small Cheeger constant, that is, a narrow bottleneck.\nA dumbbell, two round spheres joined by a thin tube, has a small Cheeger constant and also a small $\\lambda_1$.\nThe reverse bound needs more hypotheses: Buser proved an upper bound for $\\lambda_1$ in terms of $h(M)$ under a lower bound on the Ricci curvature.\nThe [[Cheeger constant of a graph|Y7S6W12N64]] satisfies a discrete analogue of the inequality.\n\n## Explanations\n\n### L1 human\n\nThe Cheeger constant measures the narrowest bottleneck of a shape: the smallest cut that splits it into two big pieces.\nThe first eigenvalue measures how fast heat evens out over the shape.\nCheeger's inequality says that a shape without narrow bottlenecks must have a large first eigenvalue.\nSo geometry, the size of cuts, controls analysis, the rate of mixing.\n\n### L2 human\n\nThe inequality gives a lower bound on $\\lambda_1$ from purely geometric data, without assumptions on curvature.\nIts discrete version for graphs is a basic tool for expander graphs, where the spectral gap and the edge expansion control each other.\nIt complements curvature bounds such as the [[Lichnerowicz–Obata theorem|TK1PT9FDV0]], which need a positive lower bound on the Ricci curvature.\n",
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  "name": "Cheeger's inequality for the first eigenvalue",
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