Hard metrics from Cayley graphs of Abelian groups

Hard metrics are the class of extremal metrics with respect to embedding into Euclidean Spaces: their distortion is as bad as it possibly gets, which is Omega(log n). Besides being very interesting objects akin to expanders and good codes, with rich structure of independent interest, such metrics are important for obtaining lower bounds in Combinatorial Optimization, e.g., on the value of MinCut/MaxFlow ratio for multicommodity flows. For more than a decade, a single family of hard metrics was known (see [10,3]). Recently, a different such family was found (see [8]), causing a certain excitement among the researchers in the area. In this paper we present another construction of hard metrics, different from [10,3], and more general yet clearer and simpler than [8]. Our results naturally extend to NEG and to l(1).