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On the Volume of Unions and Intersections of Balls in Euclidean Space

  • Y. Gordon
  • M. Meyer
Conference paper
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Abstract

We study the following old problem: Given two sequences { ai } i N =1 and {b i } i N =1 of N points in ℝ n , and positive scalars {r i } i N =1 such that |a i a j | ≤ |b i b j | for all i, j, does it follow that
$$ vo{l_n}\left({\bigcup\limits_{i = 1}^N {B\left({{a_i},{r_i}} \right)}} \right) \leqslant vo{l_n}\left({\bigcup\limits_{i = 1}^N {B\left({{b_i},{r_i}} \right)}} \right) $$
where |. | is the Euclidean norm and B(a, r) is the ball centered at a and of radius r? Under some additional assumptions, we give a probabilistic proof of this and of other related results.

Keywords

Convex Body Euclidean Norm Surface Area Measure Positive Definite Symmetric Matrix Symmetric Convex Body 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • Y. Gordon
    • 1
  • M. Meyer
    • 2
  1. 1.Technion HaifaIsrael
  2. 2.University of Paris VIParisFrance

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