Optimization of shape in continuum percolation
成果类型：
Article
署名作者：
Jonasson, J
署名单位：
University of Gothenburg; Chalmers University of Technology
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1008956687
发表日期：
2001
页码：
624-635
关键词：
摘要：
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 lambda, a copy of a given compact convex set A with fixed rotation is placed. To each A we associate a critical value lambda (c)(A) which is the infimum of intensities lambda for which the occupied component contains an unbounded connected component. It is shown that min{lambda (c)(A) : A convex of area a} is attained if A is any triangle of area a and max {lambda (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, 4 mu (A) less than or equal to mu (A circle plus (A) over tilde) less than or equal to 6 mu (A) with equality to the left iff A is centrally symmetric and with equality to the right iff A is a triangle. Here mu, denotes area and A circle plus (A) over tilde 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 mu (C) = mu (T) = mu (A) and C circle plus (C) over tilde subset of or equal to A circle plus (A) over tilde subset of or equal to T circle plus (T) over tilde with equality to the left iff A is centrally symmetric and to the right iff A is a triangle.