The Cheeger inequality provides a relationship between the spectral properties of a graph or a Riemannian manifold and its geometry, particularly focusing on how the smallest non-zero eigenvalue of the Laplacian relates to the 'cheeger constant'. This constant measures the minimum ratio of the boundary size to the volume of a subset, offering insights into the connectivity and geometric properties of the space. It connects concepts of spectral theory, geometry, and analysis.
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