Average measure of intersection of a convex region with its translate Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about 
$$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$ 
How large can $f(C)$ be? Of course, there is the trivial bound $f(C)\leq \lambda(C)^2$ but I would expect more something like $f(C) = O( \lambda(C) )$ at least. This question has probably been answered somewhere, and I suspect that the following might be very well known: for regions $C$ of constant measure, the value of $f(C)$ is maximal if $C$ is an $n$-ball (or something similar). I couldn't find any useful references regarding this question. If someone could help me out with some reference, or even a simple solution, I would be very grateful.
 A: How about a short solution to a much general problem, and without convexity assumption (which turns out to  be really just a distraction here) ?

So, more generally, for nonempty compact subsets $A$, $B$, and $C$ of $\mathbb R^n$, define $F(A,B,C)$ by
$$
F(A,B,C) := \int_{A}\lambda(B \cap (x + C))dx.
$$

In particular, the OP's question is concerned with $f(C) := F(C,C,C)$.

We shall proceed in 3 short steps.

*

*Step 1. The first step in analyzing $F$ is to realize that we can rewrite it as $F(A,B,C) = G(1_A,1_B,1_C)$, where

\begin{align}
G(u,v,w) = \int_{\mathbb R^n}\int_{\mathbb R^n} u(x)w(x-y)v(y)\,dxdy.
\end{align}

*

*Step 2. Now, by the Riez rearrangement inequality, we know that
$$
G(u,v,w) \le G(u^\star,v^\star,w^\star), $$
where $u^\star$ is the Symmetric decreasing rearrangement of the function $u$. On the other hand, it is clear that
$$
(1_A)^{\star} = 1_{K(A)},
$$
where $K(A)$ is the centered ball of same volume as $A$.
We deduce that
$$
F(A,B,C) \le F(K(A),K(B),K(C)),
$$
and in particular, $f(C) \le f(K(C))$. The solution to the OP's question then follows from computation for the ball, in the begining of the accepted answer https://mathoverflow.net/a/282941/78539.
