Optimization of Shape In Continuum Percolation

Abstract

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 $\min\{\lambda_c(A):A \text{ convex of area } a\}$ is attained if $A$ is any triangle of area $a$ and $\max\{\lambda_c(A):A \text{ convex of area } a\}$ 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 difference­body inequality for convex sets in the plane. In the plane, the difference­body inequality states that for any compact convex set $A, 4\mu (A) \le \mu (A \oplus \check{A}) \le 6\mu (A)$ 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 $A \oplus \check{A}$ is the difference­body 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 $C \oplus \check{C} \subseteq A \oplus \check{A} \subseteq T \oplus \check{T}$ with equality to the left iff $A$ is centrally symmetric and to the right iff $A$ is a triangle.