Sets and partitions minimising small differences

For a bounded measurable set $A\subseteq \mathbb{R}$ we denote the Lebesgue measure of $\{(x,y)\in A^2\text{\hspace{0.17em}⁣}:x\le y\le x+1\}$ by $\mathrm{\Phi }(A)$. We prove that if $I=A_1\cup \cdots \cup A_{k+1}$ partitions an interval $I$ of length $L$ into $k+1$ measurable pieces, then ${\sum }_{i=1}^{k+1}\mathrm{\Phi }(A_i)\ge (\sqrt{k^2+1}-k)L-1$, where the multiplicative constant $\sqrt{k^2+1}-k$ is optimal. As a matter of fact we obtain the more general result that $\mathrm{\Phi }(A)\ge (\xi +\sqrt{1-2\xi +2{\xi }^2}-1)L-1$ whenever $A\subseteq I$ has measure $\xi L$.