Note on Bessaga-Klee classification

TL;DR

This paper reviews and consolidates various proofs of the Bessaga-Klee classification of convex bodies in topological vector spaces, correcting errors and simplifying existing proofs to enhance understanding and correctness.

Contribution

It compiles multiple proof variants, corrects a known error, and simplifies a classification proof for convex bodies in topological vector spaces.

Findings

Identified an error in Bessaga and Pelczyński's original proof.

Provided a complete, corrected proof of the third case of the classification.

Simplified the proof of smooth convex bodies classification in Banach spaces.

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.~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.