We prove that the –snowflake of any finite-dimensional normed space embeds into a Hilbert space with quadratic average distortion
We deduce from this (optimal) statement that if an –vertex expander embeds with average distortion into , then necessarily , which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound 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).
"An average John theorem." Geom. Topol. 25 (4) 1631 - 1717, 2021. https://doi.org/10.2140/gt.2021.25.1631