I am quite new to the area of metric embeddings so this question might turn out to be extremely easy.
Consider a metric supported on the edges of a boolean hypercube. By supported I mean every edge of the boolean hypercube has a non negative distance associated with it and the metric is defined by the length of the shortest path according to the distance function between any two vertices. Can we put upper bounds/lower bounds on the distortion when we embed such a metric into $l_1$ ?
Any references would be highly appreciated.