Abstract
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.
Pelczynski (1975). We further provide a simplified version of T.
Dobrowolski's proof of the smooth classification of smooth convex bodies in Banach spaces which also works in the topological case.