I am searching for a result in the literature that I am sure must be known, but I just fail to find it.

Let us starts with a simple example: Let $A, B\subset \mathbb{Z}$ be a finite sets of integers such that $|A|=|B|=2m+1$ and denote by $I$ the set $\{-m,...,m\}$. I want to show that for all $k,l \geq 0$ we have $$\sum_{|i|\leq k, |j|\leq l}|(A+i)\cap(B+j)|\leq \sum_{|i|\leq k, |j|\leq l}|(I+i)\cap(I+j)|.$$

And, more generally, for any finite collection of sets $A_1,\ldots,A_t$ of the same odd cardinality $2m+1$ and all numbers $i_1,\ldots, i_t\geq 0$ we have $$\sum_{|j_1|\leq i_1,\ldots, |j_t|\leq i_t}|(A_1+j_1)\cap\ldots \cap(A_t+j_t)|\leq \sum_{j_1|\leq i_1,\ldots, |j_t|\leq i_t}|(I+j_1)\cap\ldots \cap(I+j_t)|.$$

Thank you for your attention!