A closedness criterion for the difference of two closed convex sets in general Banach spaces (Q1819226)

Let \(A\) be a non-empty, closed convex subset of a Banach space \(E\) and denote by \(\text{ca} A\) the asymptotic cone of \(A\) which is the greatest cone included in \(A\) if \(A\) contains \(0\), the origin of \(E\). The purpose of the present paper is to prove the following result (Theorem 1.1): Let \(A\) and \(B\) be two closed convex subsets of a Banach space \(E\) which satisfy the conditions that \(A\) is included in a finite dimensional subspace of \(E\) and \(\text{ca} A{\cap}\text{ca} B\) is a linear subspace of \(E\). Then the difference \(A-B\) is a closed convex subset of \(E\).