An average John theorem

Abstract

We prove that the $\frac{1}{2}$–snowflake of any finite-dimensional normed space $X$ embeds into a Hilbert space with quadratic average distortion

$$O\left(\sqrt{logdim\left(X\right)}\right).$$

We deduce from this (optimal) statement that if an $n$–vertex expander embeds with average distortion $D⩾1$ into $X$, then necessarily $dim\left(X\right)⩾n^{\mathrm{\Omega }\left(1∕D\right)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $dim\left(X\right)\gtrsim {\left(logn\right)}^2∕D^2$ of Linial, London and Rabinovich (1995), strengthens a theorem of Matoušek (1996) which resolved questions of Johnson and Lindenstrauss (1982), Bourgain (1985) and Arias-de-Reyna and Rodríguez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).