$\delta$ hyperbolic geodesic metric spaces

Question

I have a basic question about the definition of $\delta$ hyperbolic geodesic metric spaces using triangles (studied in geometric group theory cours). The definition I studied in class is that a geodesic metric space is $\delta$ hyperbolic if for every geodesic triangle with sides (a,b,c):
$d(a,b\bigcup c):=inf(d(x,y)|y\in a, x\in b\bigcup c)\leqslant\delta$.
What I don't understand is, why isn't the infimum always zero? in every triangle the sides must intersect, don't they?

Answer 1

You're right to suspect that definition for $\delta$-hyperbolicity: it is the wrong definition.

Given the geodesic triangle with sides $a,b,c$, the correct definition is not a bound on $d(a, b \cup c)$, instead:

For each $y \in a$, $d(y,b \cup c) = \inf\{d(y,x) \mid x \in b \cup c\} \le \delta$.

One could, if one so desired, write this as a "sup-inf":

$\sup_{y \in a} \bigl( \inf\{d(y,x) \mid x \in b \cup c\} \bigr) \le \delta$

(In fact this is even a "max-min", using compactness of the sides of the triangle.)