Hyperplane Arrangements with Large Average Diameter

Antoine Deza; Feng Xie

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Hyperplane Arrangements with Large Average Diameter

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arxiv 0710.0328 v1 pith:ZHCLOMBJ submitted 2007-10-01 math.MG

Hyperplane Arrangements with Large Average Diameter

Antoine Deza , Feng Xie This is my paper

classification math.MG

keywords dimensionarrangementsaveragediameterasymptoticallygivehavinghyperplane

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The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed dimension. We give the exact value of the largest possible average diameter for all simple arrangements in dimension 2, for arrangements having at most the dimension plus 2 hyperplanes, and for arrangements having 6 hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds which are both asymptotically equal to the dimension.

Hyperplane Arrangements with Large Average Diameter

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