inequalities – Shrinking subset and product


Given a segment and a value $c$ less than the segment length, let $A_1,dots,A_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|Bcap A_1|timesdotstimes |Bcap A_n|$, where $|cdot|$ denotes the length (i.e. Lebesgue measure). If there are many such $B$, we choose one arbitrarily.

Now, we shrink $A_1$ to $A_1’subseteq A_1$, and choose $B’$ using the same procedure. Is it always true that $|B’cap A_1’|le |Bcap A_1|$?

If $A_1,dots,A_n$ are disjoint finite unions, the answer is positive, as shown here.