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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 \ldots \le t_n = 1} \sum_i d(\gamma(t_i),\gamma(t_{i+1})