Optimization of shape in continuum percolation
Optimization of shape in continuum percolation
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DOI：
10.1214/aop/1008956687
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发表时间：
2001-04
影响因子：
2.3
通讯作者：
J. Jonasson
中科院分区：
文献类型：
--
作者：
J. Jonasson
We consider a version of the Boolean (or Poisson blob) continuum percolation model where, at each point of a Poisson point process in the Euclidean plane with intensity A, a copy of a given compact convex set A with fixed rotation is placed. To each A we associate a critical value λ c (A) which is the infimum of intensities A for which the occupied component contains an unbounded connected component. It is shown that min{λ c (A): A convex of area a} is attained if A is any triangle of area a and max{λ c (A): A convex of area a} is attained for some centrally symmetric convex set A of area a. It turns out that the key result, which is also of independent interest, is a strong version of the difference-body inequality for convex sets in the plane. In the plane, the difference-body inequality states that for any compact convex set A, 4μ(A) ≤ < μ(A ○+ A) ≤ 6μ(A) with equality to the left iff A is centrally symmetric and with equality to the right iff A is a triangle. Here μ denotes area and A ○+ A is the difference-body of A. We strengthen this to the following result: For any compact convex set A there exist a centrally symmetric convex set C and a triangle T such that μ(C) = μ(T) = μ(A) and C ○+ C ⊆ A ○+ Ă ⊆ T ○+ T with equality to the left iff A is centrally symmetric and to the right iff A is a triangle.
