For my use case, it seems that $L_1$ distance on simplex
$$\lVert x - y \rVert_1 = \sum |x_i - y_i|$$
is more useful when written in the following form:
$$\lVert x - y\rVert_1 = 2 - 2\sum \min(x_i, y_i)$$
(very simple proof left to the reader). This reminds me somewhat of the dual formulation of Wasserstein distance, where we equate a distance based on a minimum with one based on a maximum. Is what I wrote out an instance of a similar underlying concept, or any more general concept? Or is it just a one-off relationship?