Regularity of shadows and the geometry of the singular set associated to a Monge-Ampere equation

2013/11/22 by Emanuel Indrei, Indrei, Emanuel, Levon Nurbekyan +1
Mathematics · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC) #math.AP #math.CA #math.FA #math.OC

paper · pdf · doi:10.48550/arxiv.1311.5865

arxiv created 2013/11/22 · arxiv updated 2013/11/25

Abstract

Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region. We prove sharp regularity results for the boundary of this shadow in every direction of illumination. Moreover, techniques are developed for investigating the regularity of the region generated by orthogonally projecting a convex set onto another. As an application we study the geometry and Hausdorff dimension of the singular set corresponding to a Monge-Ampere equation.

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