Convexity and Intersection of Random Spaces

  • Mariya Shcherbina
  • Brunello Tirozzi
Part of the Developments in Mathematics book series (DEVM, volume 8)


The problem of finding the volume of the intersection of the N dimensional sphere with p = αN random half spaces when α is less than a critical value α c , and when N,p → ∞is solved rigorously. The asymptotic expression coincides with the one found by E. Gardner ([4]), using non rigorous replica calculations in neural network theory. When α is larger than α c the volume of the intersection goes to 0 more rapidly than exp(— N const). We use the cavity method. The convexity of the volume and the Brunn Minkowski theorem ([3]) have a central role in the proof.


Random systems Convexity 


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Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Mariya Shcherbina
    • 1
  • Brunello Tirozzi
    • 2
  1. 1.Institute for Low TemperaturesUkr. Ak. SciKharkovUkraine
  2. 2.Department of PhysicsUniversity of Rome “La Sapienza”RomaItaly

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