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A Minkowski-style theorem for focal functions of compact convex reflectors

Author: Vladimir I. Oliker
Journal: Trans. Amer. Math. Soc. 360 (2008), 563-574
MSC (2000): Primary 49K20, 35J65, 78A05
Published electronically: September 21, 2007
MathSciNet review: 2346462
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Abstract: This paper continues the study of a class of compact convex hypersurfaces in $ \mathbf{R}^{n+1}, ~n \geq 1,$ which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $ \mathbf{R}^3$ reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are also important in rendering problems in computer graphics.

The notion of a focal function for reflectors plays a central role similar to that of the Minkowski support function for convex bodies. In this paper the basic question of when a given function is a focal function of a convex reflector is answered by establishing necessary and sufficient conditions. In addition, some smoothness properties of reflectors and of the associated directrix hypersurfaces are also etablished.

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Additional Information

Vladimir I. Oliker
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322-0239

Received by editor(s): December 1, 2004
Published electronically: September 21, 2007
Additional Notes: The research of the author was partially supported by the National Science Foundation grant DMS-04-05622, the Air Force Office of Scientific Research under contract FA9550-05-C-0058 and by a grant from the Emory University Research Committee.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.