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

TL;DR

This paper introduces an open problem in the theory of diversities, focusing on the worst-case minimal distortion embedding of diversities into , highlighting its relation to classical metric space problems and discussing potential research directions.

Contribution

It formulates the open problem, compares it to known metric space results, and discusses why existing techniques are insufficient, proposing new avenues for investigation.

Findings

Identified special classes of diversities with known embeddings

Explained limitations of standard metric techniques for diversities

Outlined potential approaches for future research

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.