An average John theorem

Abstract

We prove that the $\frac{1}{2}$-snowflake of a finite-dimensional normed space $(X,\Vert \cdot {\Vert }_X)$ embeds into a Hilbert space with quadratic average distortion $$O\left(\sqrt{\log \dim (X)}\right).$$ We deduce from this (optimal) statement that if an $n$-vertex expander embeds with average distortion $D⩾1$ into $(X,\Vert \cdot {\Vert }_X)$, then necessarily $\dim (X)⩾n^{\Omega (1/D)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $\dim (X)\gtrsim (\log n{)}^2/D^2$ 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 Rodr{\'ı}guez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).

arXiv commentReferee comments addressed. To appear in Geometry & Topology

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