  We collect several variants of the proof of the third case of the
Bessaga-Klee relative classification of closed convex bodies in topological
vector spaces. We were motivated by the fact that we have not found anywhere in
the literature a complete correct proof. In particular, we point out an error
in the proof given in the book of C.~Bessaga and A.~Pe\l czy\'nski (1975). We
further provide a simplified version of T.~Dobrowolski's proof of the smooth
classification of smooth convex bodies in Banach spaces which works
simultaneously in the topological case.
