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Random Polytopes

Rajaee, Mohaddeseh | 2011

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 44607 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Alishahi, Kasra
  7. Abstract:
  8. Random Polytopes, the first occurrence of which dates back to the famous Sylvester’s four points problem in the 1860s, is a branch of geometric probability, typically concerning the convex hull of some random points chosen from a convex subset of Rd. In this thesis we have studied some special kind of random polytopes; the one that is the convex hull of some independent random points chosen from a convex body (a convex, compact set with interior point) according to the uniform distribution. It was a new approach from A. Rényi and R. Sulanke in 1963 to consider this type when the number of random points tends to infinity.This thesis consists of three main parts: The first part is devoted to the order of E(K, n) and E(fi(Kn)). Here, K is a convex body with volume one from which the points are chosen, n is the number of random points, E(K, n) is the expected value of missed volume, and E(fi(Kn)) is the expected number of i-dimensional faces of the random polytope Kn. The most important fact in this part is a theorem due to I. Bárány and D. Larman which asserts that E(K, n) has the same order as the volume of the wet part of K with parameter 1/n. Finding upper and lower bounds for the order of E(K, n) has been based on this fact. Then, we have introduced classes of convex bodies, the upper and lower bounds are attained for. The boundary structure of K is important here. Finally, using the Efron’s identity we have calculated the order of E(fi(Kn)). In the second part, it has been shown that for a typical convex body K of volume one (in the sense of Baire’s category), there is no exact order of E(K, n). In fact, for such elements, E(K, n) is infinitely many times larger than any order less than the upper bound and infinitely many times smaller than any order larger than the lower bound. Here, the main tool is a theorem proven by P. Gruber. In the third part, we have given a sketch of proof for the fact that the sequence of missed volumes, as a sequence of random variables, satisfies a central limit theorem when K is a polytope. At the end, there are two appendices. The first one is devoted to convex geometry and the other contains the proof of a lemma from part three
  9. Keywords:
  10. Geometric Probability ; Random Polytopes ; Economic Cap Covering Theorem ; Irregularity Criterion ; Central Limit Theorem ; Convex Bodies Approximation

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