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Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following: It's written in wiki in examples of CAT($k$) spaces (https://en.wikipedia.org/wiki/CAT(k)_space) that the closed subspace $X$ of $\mathbb E^3$ (where $\mathbb E^3$ is a 3-dimensional Euclidean space) given by $X=\mathbb E^3 \setminus \{(x,y,z)|x>0, y>0, z>0\}$ equipped with the induced length metric is not a CAT($k$) space for any $k$. There was the following advice in comment of MSE: consider triangle with vertices on the border: $(x,0,0)$, $(0,y,0)$, $(0,0,z)$. In this case as I understand I need to use CAT($k$) inequality and the following proposition: geodesic metric space $X$ is said to be CAT($k$) space if every geodesic triangle in $X$ with perimeter less then $2D_k$ satisfies the CAT($k$) inequality (Where $D_k$ is a diameter). If I'm right why we have geodesic triangle in $X$ with perimeter less than $2D_k$ which is not satisfied by this CAT($k$) inequality?

I want to understand how it all works.

If you don't mind can you please explain why it's not a CAT($k$) space in more details?

Thank you!

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    $\begingroup$ The triangle $\triangle$ with the vertices you list can be chosen to have small diameter. Consider the comparison triangle $\tilde\triangle$ in the curvature $k$ model plane (a sphere if $k>0$) and compare the corresponding angles of $\triangle$ and $\tilde\triangle$. $\endgroup$
    – Igor Belegradek
    Commented Jun 5, 2022 at 20:52
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    $\begingroup$ By the way, it is easy to prove that if a polyhedral space is CAT(к), then it is CAT(0). So you should not worry about к. $\endgroup$
    – Anton Petrunin
    Commented Jun 5, 2022 at 21:33
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    $\begingroup$ The statement about Banach space is not related. It follows since the CAT-comparison implies the parallelogram identity. $\endgroup$
    – Anton Petrunin
    Commented Jun 5, 2022 at 21:49
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    $\begingroup$ The space contains arbitrarily small geodesic bigons, which is impossible for a CAT(k) space. $\endgroup$
    – Moishe Kohan
    Commented Jun 5, 2022 at 22:31
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    $\begingroup$ As I said you need to read a textbook. If I may, I suggest Burago-Burago-Ivanov as a most friendly exposition. Tutors are not needed but it will be a substantial time investment. At this point you don't seem to know basis definitions, which is where your questions originate. $\endgroup$
    – Igor Belegradek
    Commented Jun 5, 2022 at 22:53

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