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Discrete & Computational Geometry

, Volume 47, Issue 2, pp 275–287 | Cite as

Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

  • Károly Bezdek
Article

Abstract

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating convex bodies of constant width to the family of “fat” spindle convex bodies. Also, this leads to the spherical analog of the well-known Blaschke–Lebesgue problem.

Keywords

Boltyanski–Hadwiger illumination conjecture (Fat) spindle convex bodies Spindle convex hull Gauss (resp., normal) images of faces Illumination by random directions Convex bodies of constant width in spherical space 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of PannoniaVeszprémHungary
  3. 3.Institute of MathematicsEötvös UniversityBudapestHungary

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