Let H be an infinite dimensional and separable Hilbert Space. Do there exist disjoint, closed and bounded subsets A,B of H which satisfy the following conditions? (1) Each of A,B is convex and has a non-empty interior with respect to H. (2) Given any positive real number e, there is a point of A and a point of B whose distance apart is less than e....If the answer to this question is "YES", can A and B be congruent?...Any information about this subject will be much appreciated.



Two questions about convex subsets of Hilbert Space
No. Convex closed subsets are weakly closed, and since $A,B$ are also assumed bounded, we can make $x\_n,y\_n$ converge weakly to limits $x\in A$, $y\in B$ on a subsequence if $x\_n\in A$, $y\_n\in B$. If now also $\|x\_n -y\_n\|\to 0$, then $x=y\in A\cap B$.


