The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {Dn:n ∈N}, a non-decreasing family of non empty, closed, convex and bounded sets such that Dn+Dn+2⊆2Dn+1for every n ∈N. The second part of this article proves that in every infinite dimensional real Banach space there is a convex set which can be expressed as the union of countably many closed sets, but not as the union of countably many closed and convex sets. Accordingly, every infinite dimensional real Banach space contains a convex Fσset which is not the image of a closed convex set under a Fredholm operator.