Geometriae Dedicata

, Volume 80, Issue 1, pp 319–329

On Translational Clouds for a Convex Body

  • István Talata

DOI: 10.1023/A:1005279901749

Cite this article as:
Talata, I. Geometriae Dedicata (2000) 80: 319. doi:10.1023/A:1005279901749


For a d-dimensional convex body K let C(K) denote the minimum size of translational clouds for K. That is, C(K) is the minimum number of mutually non-overlapping translates of K which do not overlap K and block all the light rays emanating from any point of K. In this paper we prove the general upper bound \(C(K) \leqslant 6^{d^2 + o(d^2 )}\). Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show \(C(S) \leqslant 3^{d^2 + o(d^2 )}\). Finally, for the d-dimensional ball Bd we obtain the bounds \(2^{0.599d^2 - o(d^2 )} \leqslant C(B^d ) \leqslant 2^{1.401d^2 + o(d^2 )}\).

convex body convex cone cloud covering difference body translative packing 

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • István Talata
    • 1
  1. 1.Department of MathematicsAuburn UniversityAuburnU.S.A.

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