Is $\int_{-c}^c |A \cap (x + A)|\, dx$ maximized when the measurable subset $A \subseteq \mathbb R$ is an interval centered at the origin?

Question

Let $A$ be a nonempty measurable subset of $\mathbb R$, with Lebesgue measure $|A|=1$, and let $c>0$. Define the scalar $I(A)$ by

$$ I(A) := \int_{-c}^c |A \cap (x + A)|\, dx, $$ where $x+A := \{x + a \mid b \in A\}$.

Question. Under the constraint $|A| = 1$, is it true that $I(A)$ is maximized when $A$ is an interval centered at the origin, i.e $[-1/2,1/2]$ ?

In the case where $A$ is restricted to an interval, then the answer to the question is affirmative, and has been provided here https://mathoverflow.net/a/282941/78539 (under the name "Fedja's lemma"). Unfortunately, the solution is somewhat complicate and I don't see how to modify it so that it applies without convexity (i.e non-interval sets).

Answer 1

Disclaimer: Upon recommendation (in the comments), this post is just to close off the question, since it has been completely answered in the comments.

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• Indeed, as observed by user @Terry Tao in the comments, the question has a simple affirmative answer via the Riez rearrangment inequality.

• Using this machinery, a more general question in $n$ dimensions is answered here https://mathoverflow.net/a/415754/78539, and the answer to my original question (on the real line) follows as an immediate corollary.