# 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 $$|B\cap A_1|\times\dots\times |B\cap A_n|$$, 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 $$|B'\cap A_1'|\le |B\cap A_1|$$?

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

$$\newcommand\om\omega\newcommand\Om\Omega\newcommand\de\delta$$As in the linked answer, this problem on sets can be restated as the following problem on real numbers.

Let $$\Om:=\{0,1\}^n$$. For each $$\om\in\Om$$ and each $$i\in[n]:=\{1,\dots,n\}$$, let $$\om_i$$ denote the $$i$$th coordinate of the vector $$\om$$, so that $$\om=(\om_1,\dots,\om_n)$$.

For any $$a=(a^\om)\in[0,\infty)^\Om$$ and any $$c\in[0,\sum_{\om\in\Om} a^\om]$$, let $$B(a):=B_c(a):=\Big\{b=(b^\om)\in[0,\infty)^\Om\colon0\le b^\om\le a^\om\ \forall\om\in\Om,\sum_{\om\in\Om\setminus\{(0,\dots,0)\}}b^\om=c\Big\}.$$

Let $$b(a)=b_c(a)$$ be any maximizer of $$\pi(b):=\prod_{i=1}^n\sum_{\om\in\Om\colon\,\om_i=1}b^\om$$ over all $$b\in B(a)$$.

Take now any $$u=(u^\om)\in[0,\infty)^\Om$$ such that $$u^\om\le a^\om$$ for all $$\om\in\Om$$ with $$\om_1=1$$ and $$u^\om=a^\om$$ for all $$\om\in\Om$$ with $$\om_1=0$$. Let $$b(u)=b_c(u)$$ be any maximizer of $$\pi(b)$$ over all $$b\in B(u)$$. Does it then necessarily follow that $$\sum_{\om\in\Om\colon\,\om_i=1}b(u)^\om\le\sum_{\om\in\Om\colon\,\om_i=1}b(a)^\om?$$

In this restatement, for all $$\om\in\Om$$

• $$a^\om$$ stands for $$|A^\om|$$, where $$A^\om:=A^{\om_1}\cap\cdots\cap A^{\om_n}$$ and, for $$\de\in\{0,1\}$$ and $$A\subseteq J$$, where $$J$$ is the segment in question, we let $$A^\de:=A$$ if $$\de=1$$ and $$A^\de:=J\setminus A$$ if $$\de=0$$;
• $$b^\om$$ stands for $$|B\cap A^\om|$$.

We see that even the restated problem, in terms of real numbers, concerns maximizing the non-convex/non-concave function $$\pi(\cdot)$$ over a polytope of dimension $$2^n-1$$, defined by $$2^{n+1}$$ affine inequalities. Moreover, we need to compare solutions of two such problems. This seems hard! If someone can answer this question, it could be quite an instructive moment!