Communications in Analysis and Geometry

Volume 24 (2016)

Number 4

Regularity of shadows and the geometry of the singular set associated to a Monge–Ampère equation

Pages: 793 – 820



E. Indrei (Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

L. Nurbekyan (CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia)


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 domain onto another. As an application we study the geometry and Hausdorff dimension of the singular set corresponding to a Monge–Ampère equation.

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