mean a path with same initial point and terminal point.

A loop $\gamma:[a,b]\rightarrow X$ is simple if $\gamma(t_1)\neq \gamma(t_2)$ for $t_1\neq t_2$ in $[a,b)$.

\[corollary 4.18, [@DS05b]\]\[limitgeodesictrg\] Assume that in an asymptotic cone $Con^\omega(X,e,d)$, a collection of closed subsets $\mathcal{P}$ satisfying ($T_1$) and every non trivial simple geodesic triangle in $Con^\omega(X,e,d)$ whose sides are limit geodesic is contained in a subset from the collection $\mathcal{P}$. Then $\mathcal{P}$ satisfy ($T_2$) i.e, $Con^\omega(X,e,d)$ is tree-graded with respect to $\mathcal{P}$.

\[asymptoticgromov\] A geodesic metric space is hyperbolic if and only if all of its asymptotic cones are $\mathbb{R}$-tree.

Hyperbolic group and hyperbolicity function
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Let $(X,d)$ be a geodesic metric space. For every positive real number $t$, define the following set: $$S_t:=\{ set \ of \ all \ geodesic \ triangles \ with \ perimeter \ \leq t \}$$ For a geodesic triangle triangle $\triangle$ with sides $A,B,C$ in $X$ define, $$\delta_{\triangle}:= \underset{p\in A\cup B\cup C}{\sup}d(p,union \ of \ two\ sides\ of\ \triangle \ not \ containing \ p)$$ Define the hyperbolicity function $f_X:[0,\infty)\rightarrow [0,\infty)$ by $f_X(t):=\underset{\triangle \in S_t}{\sup}\delta_{\triangle}$.

One can reformulate lemma $\color{blue}{\ref*{hyperbolicngon}}$ for a geodesic metric space in terms of it’s hyperbolicity function as follows,

\[hyperbolicgeodesicngon1\] For $n\geq 3$, any side 