In finite dimensional normed space, every convex set contains a basis

Question

I've been reading Lemma 5.60 here: http://epge.fgv.br/we/MD/TeoriaEconomicaAvancadaI/2009?action=AttachFile&do=get&target=Aliprantis-Infinite-Dimensional-Analysis.pdf (p.g 200, =217 on the pdf)

And couldn't understand the very beggining of the proof. Why such basis should exist?

Also, if anyone knows a nicer proof for this lemma i'd love to hear it.

Thanks

Please don't mistake internal to be interior. The definition of internal points is in the top of the page.

Answer 1

The proof is acting in a sloppy way. It should say, "Let $C$ be a convex set containing at least one internal point" (if it contains no internal points, the implication is vacuous). Then, the fact that such a $C$ contains a basis is simple! Choose an internal $c\in C$. Then, for some $\alpha$, the vectors $c+\alpha e_i\in C$ for a basis $\{e_1,\ldots,e_n\}$. So it is then clear that $C$ contains a basis.