Given a metric space $X$ and an equivalence relation $\sim$, the quotient (pseudo-)metric on $X/\sim$ is defined as follows:
- $d'([x],[y]) = \inf \left \{ d(p_1,q_1) + d(p_2,q_2) + ... + d(p_n,q_n)\right \}$
where $[p_1] = [x], [q_n] = [y],$ and $[q_i] = [p_{i+1}]$.
Under what circumstances does this reduce to the much simpler equation
- $d'([x],[y]) = \inf \left \{ d(p,q): [p] = [x], [q] = [y] \right \}$
?
It appears to work for $\mathbb{R}^n/\mathbb{Z}^m$ for $m \leq n$, but I'm curious about the general case.