I will show the first approximate nearest neighbor search data structure for a general d-dimensional normed space with sub-polynomial in d approximation.

The main tool is a finite-dimensional quantitative version of a theorem of Daher, which yields a Holder homeomorphism between small perturbations of a normed space of interest and a Euclidean space. To make Daher's theorem algorithmic, we employ convex programming to compute the norm of a vector in a space, which is the result of complex interpolation between two given normed spaces.

Based on a joint work (FOCS 2018) with Alex Andoni, Assaf Naor, Sasho Nikolov and Erik Waingarten.