I have another new paper up on arXiv, titled "Manhattan orbifolds". It's about 2-dimensional manifolds with hyperconvex metrics, which is to say that they have properties similar to $$L_\infty$$ metrics in $$\mathbb{R}^n$$.

The original idea was that these things might be useful in designing approximation algorithms for the hyperbolic plane, related to my squarepants paper. But, though the paper does describe a hyperconvex approximation to the hyperbolic plane, I haven't yet found any approximation algorithms based on it, and there isn't really any algorithmic content in the paper. Instead it turned into something about distances in graphs (squaregraphs, which are a type of partial cube, and certain other related graphs).