Constant Distortion Embeddings of Symmetric Diversities

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into L-1, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of L-1 spaces. In the metric case, it is well known that an n-point metric space can be embedded into L-1 with O (log n) distortion. For diversities, the optimal distortion is unknown. Here, we establish the surprising result that symmetric diversities, those in which the diversity (value) assigned to a set depends only on its cardinality, can be embedded in L-1 with constant distortion.