# An average John theorem

## TL;DR

This paper proves that the 1/2-snowflake of finite-dimensional normed spaces embeds into Hilbert space with low average distortion, leading to sharp bounds on the dimension needed for embedding expanders, improving previous results.

## Contribution

It establishes an optimal embedding result for snowflakes of normed spaces and derives sharp dimension bounds for embedding expanders, resolving longstanding open questions.

## Key findings

- The 1/2-snowflake of finite-dimensional normed spaces embeds into Hilbert space with distortion O(√log(dim))
- Dimension bounds for embedding expanders into normed spaces are sharp and improve previous estimates
- Answers negatively to a question on embedding dimensions posed for algorithmic applications

## Abstract

We prove that the $\frac12$-snowflake of a finite-dimensional normed space $(X,\|\cdot\|_X)$ embeds into a Hilbert space with quadratic average distortion $$O\Big(\sqrt{\log \mathrm{dim}(X)}\Big).$$ We deduce from this (optimal) statement that if an $n$-vertex expander embeds with average distortion $D\geqslant 1$ into $(X,\|\cdot\|_X)$, then necessarily $\mathrm{dim}(X)\geqslant n^{\Omega(1/D)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $\mathrm{dim}(X)\gtrsim (\log n)^2/D^2$ of Linial, London and Rabinovich (1995), strengthens a theorem of Matou\v{s}ek (1996) which resolved questions of Johnson and Lindenstrauss (1982), Bourgain (1985) and Arias-de-Reyna and Rodr{\'{\i}}guez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).