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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A Minkowski-style theorem for focal functions of compact convex reflectors
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by Vladimir I. Oliker PDF
Trans. Amer. Math. Soc. 360 (2008), 563-574 Request permission

Abstract:

This paper continues the study of a class of compact convex hypersurfaces in $\mathbf {R}^n, ~n \geq 1,$ which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $\mathbb {R}^3$ 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
  • Email: oliker@mathcs.emory.edu
  • 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.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 563-574
  • MSC (2000): Primary 49K20, 35J65, 78A05
  • DOI: https://doi.org/10.1090/S0002-9947-07-04569-2
  • MathSciNet review: 2346462