@Benoît: Thank you, sorry about that.

Thank you very much, I was not familiar with the Toponogov's hinge comparison, I will look into that. In fact, this topic of metric geometry is all new to me, I am working with operator spaces and I found a space of finite dimensional operator spaces which is geodesic and it followed naturally the question about cat(k). If I do not ask too much, could you please answer another question? We have established that this Busemann space is not cat(k). The geodesics vary continously with the end points. Can we say more about this space? Would there be another direction I could go into?

Yes, I know, hence my question. A space which is not cat(0) but it's Busemann convex cat be cat(1)?

Misha: if X is cat(k) it is also cat(k') if k'>k. cat(0) implies cat(1) but the other implication does not seem obvious, does it?

Thank you, yes, maybe, I have this space which is for sure Busemann and NOT CAT(0) but it seems to be CAT(1) which feels a bit weird to me.

My space is not CAT(0).