Asymptotic geometry of convex sets

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My talk on limiting geometry of convex sets (inspired by Itai Benyamini's question) at Polytechnic-NYU in May 2011

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Asymptotic geometry of convex sets

  1. 1. AsymptoticGeometry of ConvexSets { Igor Rivin (Temple and IAS)
  2. 2.  Itai Benjamini and Gadi Kosma had asked whether there is a hyperbolic analogue of Dvoretzky’s TheoremThe motivating question
  3. 3.  (A. Dvoretzky, 1961): For any centrally symmetric convex body in RN there exists an almost spherical central section (this is often stated in terms of Banach spaces), almost spherical meaning that there is an inscribed and a circumscribed ball, with ratio of radii close to 1.Dvoretzky’s theorem
  4. 4.  What would a hyperbolic Dvoretzky theorem say?Hyperbolic Dvoretzky?
  5. 5.  Simpler question: what do convex bodies in hyperbolic space look like?Hyperbolic Dvoretzky
  6. 6. Hyperbolic plane
  7. 7. Hyperbolic Space
  8. 8. A counterexample?
  9. 9. Is the volume finite?
  10. 10.  Is there any relationship between the “set at infinity” and its volume?The next problem
  11. 11.  If the body intersects the ideal boundary in an open set, it has infinite volume!Obvious answer: YES
  12. 12.  (does not help us with our example…)Not quite satisfying
  13. 13.  Hyperplane in Hn intersects infinity in a set of codimension 1, has 0 volume. Not interesting…Stupid example
  14. 14.  We call a (convex) set proper, if its volume is positive and finite.Finally, a definition…
  15. 15.  The limit set C∞ of a convex set C is the intersection of C with the ideal boundary of Hn.And another…
  16. 16.  For any proper convex set in Hn,  dim C∞ ≤(n-1)/2,  Where the dimension is the upper Minkowski dimension (which upper- bounds the Hausdorff dimension).And a Theorem
  17. 17. If C∞ is smooth, then the volume of the convex hull C of C∞ is not greater than the floor of  n/2-1.And another
  18. 18.  Finite area if (and only if) the limit set C∞ is finite.Dimension 2
  19. 19.  The exist subsets of the two-sphere of arbitrary Hausdorff dimension smaller than 1, such that the volume of their convex hull is finite (based on generalized Sierpinski gaskets).  Open question: can you do dimension equal to 1?Dimension 3
  20. 20.  Fixing dimension not exceeding the “critical value” ((n-1)/2), one can always find a plane of that dimension such that the intersection of the plane and the body is bounded in terms of the volume and the dimension, and the inradius.Non-asymptoticconsequences

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