Power of weak v/s strong triangle inequalities

In a recent paper, Lee and Moharrami [13] construct a family of metrics (Xn, dn) with |Xn| = N , such that (Xn, √ dn) embeds into L2 with constant distortion, but embedding (Xn, dn) into a metric of negative type requires distortion Ω̃(log 1N). In this paper, we build on their analysis, and improve their result by showing an Ω̃(log 1N) lower bound for embedding (Xn, dn) into a metric of negative type. Moreover, we show that this analysis is essentially tight by constructing a map that has distortion O(log 1N). This result implies a lower bound of Ω̃(log 1N) for the integrality gap of the relaxed version of Goemans-Linial semidefinite program with weak triangle inequalities for Non-uniform Sparsest Cut.