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Discrete and Computational Geometry, WS1516 Exercise Sheet “10”: Last Exercise

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Discrete and Computational Geometry, WS1516 Exercise Sheet “10”: Last Exercise

University of Bonn, Department of Computer Science I

• Written solutions have to be prepared until Friday 5th of Frebuary, 12:00 pm.

• There is a letterbox in front of Room E.01 in the LBH builiding.

• You may work in groups of at most two participants.

Exercise 17: Cyclic Polytopes (4 Points)

LetV be a finite subset of the moment curve. Please prove that all the points of V are vertices of conv(V). (Hint: each of them contributes to at least one facet)

Exercise 18: Probabilistic Tools (4 Points) Let H a set of n element, and R and T be two random subsets ofH, where

|R|=r, |T|=k, andk < r < n. Please prove the followings

• The probability that R∩T 6=∅ isO(kr/n).

• If n >2r, the probability that R∩T =∅is O(e−knr)

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