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.