A general discrete Wirtinger inequality and spectra of discrete Laplacians

摘要: We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of Wirtinger inequality. It is equivalent to estimate on spectral gap a weighted Laplacian circle. The proof uses geometric construction related isoperimetric problem surface cone. In higher dimensions, mixed volumes theory leads similar results, which allows us associate Laplace operator every geodesic triangulation sphere and, by analogy, triangulated spherical cone-metric. For cone-metric with positive singular curvatures, we conjecture Lichnerowicz-Obata theorem.