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Séminaire Bourbaki

# Alexandros Eskenazis — Average distortion embeddings, nonlinear spectral gaps, and a metric John theorem after Assaf Naor

IHP
Hermite

In this lecture we shall discuss some geometric applications of the theory of nonlinear spectral gaps. Most notably, we will present a proof of a deep theorem of Naor asserting that for any norm $$\|\cdot\|$$ on $$\mathbf{R}^d$$, the metric space $$(\mathbf{R}^d, \sqrt{\|x-y\|})$$ embeds into Hilbert space with quadratic average distortion $$O(\sqrt{\log d})$$. As a consequence, we will deduce that any n-vertex expander graph does not admit a $$O(1)$$-average distortion embedding into any $$n^{o(1)}$$-dimensional normed space.