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Publication 67

On the Stability of the Polygonal Isoperimetric Inequality


CMUE. Indrei
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh PA 15213-3890 USA

Levon Nurbekyan
Center for Mathematical Analysis,
Geometry, and Dynamical Systems
Departamento de Matematica
Instituto Superior Tecnico
Lisboa 1049-001, Portugal

We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices.
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