Affine straight lines in family of bounded closed convex sets (Q2569854)

We are given a topological vector space \(X\) over the field \(R\) and \(B(X)\) a family of all nonempty bounded closed convex subsets of \(X\). Let for \(A,B \in B(X)\) the straight line in \(B(X)\) be defined as \(\varphi_{A,B}(t) = (1-t)A + tB\). The authors show in their paper that \(\varphi_{A,B}\) is not injective if and only if \(A-B\) is symmetric and \(A-A\) as well as \(B-B\) are homothetic.