Metric structures in L ₁: Dimension, snowflakes, and average distortion

We study the metric properties of finite subsets of L ₁. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L ₁. We present some new observations concerning the relation of L ₁ to dimension, topology, and Euclidean distortion. We show that every n-point subset of L ₁ embeds into L ₂ with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in L _p for p ∈(1,2). We resolve a question left open in [1] about the impossibility of linear dimension reduction in the above cases, and we show that the example of [2,3] cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.