Open Problem Statement: Minimal Distortion Embeddings of Diversities in $\ell_1$

Abstract

We state an open problem in the theory of diversities: what is the worst case minimal distortion embedding of a diversity on $n$ points in $\ell_1$. This problem is the diversity analogue of a famous problem in metric geometry: what is the worst case minimal distortion embedding of an $n$-point metric space in $\ell_1$. We explain the problem, state some special classes of diversities for which the answer is known, and show why the standard techniques from the metric space case do not work. We then outline some possible lines of attack for the problem that are not yet fully explored.