Minimizing geodesic

Definition

Given data

A metric space $M$ equipped with a metric $d$.

Definition part

A path $\gamma :[0,1]\to M$ is termed a minimizing geodesic if it is the shortest path from $\gamma (0)$ to $\gamma (1)$.

By shortest, we mean path of minimum length where the length of a path is defined as:

$\lim \sup _{0=t_{0}\leq t_{1}\leq t_{2}\leq \ldots \leq t_{n}=1}\sum _{i}d(\gamma (t_{i}),\gamma (t_{i+1})$