We establish a generic reduction from nonlinear spectral gaps of metric spaces to space partitions, in the form of data-dependent Locality-Sensitive Hashing. This yields a new approach to the high-dimensional Approximate Near Neighbor Search problem (ANN). Using this reduction, we obtain a new ANN data structure under an arbitrary d-dimensional norm, where the query algorithm makes only a sublinear number of probes into the data structure. Most importantly, the new data structure achieves a O(log d) approximation for an arbitrary norm. The only other such generic approach, via John's ellipsoid, would achieve square-root-d approximation only. Joint work with Assaf Naor, Aleksandar Nikolov, Ilya Razenshteyn, and Erik Waingarten.