Given a symmetric convex body $K \subset \mathbb{R}^n$ (i.e., a bounded symmetric convex set with non-empty interior), I am interested in the following quantity $$p_K := \Pr_{x_1, x_2 \sim K}[x_1 \in K + x_2] \; ,$$ i.e., the probability given two uniformly random and independent vectors $x_1,x_2$ in $K$ that $x_1 \in K + x_2$, which you might call the "kissing probability" of $K$, in analogy with the kissing number. (I know it's not a great name.) An equivalent definition is $$ p_K := \mathop{\mathbb{E}}_{x \sim K}[\mathrm{vol}(K \cap (K+x))/\mathrm{vol}(K)] \; . $$
For example if $K$ is the $n$-ball, then $p_K \approx (3/4)^{n/2}$, and if $K$ is the $n$-cube, then $p_K = (3/4)^n$.
My main question is whether this quantity is maximized by the ball. More generally, has $p_K$ been studied? Can we get a decent upper bound on $p_K$?
Some context:
This quantity arises relatively naturally in the study of sieving algorithms for lattice problems. At a very high level, these algorithms work by sampling a very large number of more-or-less random vectors from a convex body $K$ and then looking for pairs of vectors $x_1, x_2$ such that $x_1 \in x_2 + K/(1+\varepsilon)$, taking their difference, and repeating the procedure on the differences. The running time of these algorithms is more-or-less governed by the number of points that need to be sampled in order to guarantee that nearly every point can be paired with another, which is more-or-less $1/p_K$.
So, if one can find a convex body with $p_K \gg (3/4)^{n/2}$, then one might hope to find a faster algorithm for lattice problems, which would be quite significant.
