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:

$lim{sup}_{0=t_0\le t_1\le t_2\le \dots \le t_n=1}{\sum }_{i}d(\gamma (t_i),\gamma (t_{i+1})$