Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does there exist a function $f:U\times U\rightarrow U$ such that for any $A,B\in U$:
(a) $f(A,B) = f(B,A)$
(b) $f(A,B)$ has length (i.e. Lebesgue measure) less than $0.0001$.
(c) $f(A,B)\cap A$ has positive length.
(d) The length of $f(X,B)\cap A$ is maximized at $X=A$.
This is a variant of this question with more restrictive conditions, so my guess would be that the answer is no. Yet this variant still seems difficult.
