# 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.