A counterexample to a strengthening of a question of Milman

TL;DR

This paper constructs a high-dimensional normed space that is nearly Euclidean but lacks small subspaces that are both almost Euclidean and well complemented, challenging a strengthened version of a question by Milman.

Contribution

It provides a counterexample to a strengthened form of Milman's question, showing such nearly Euclidean and complemented subspaces may not exist.

Findings

Constructed a high-dimensional space that is strongly 2-Euclidean.

Showed this space contains no 2D subspace that is both nearly Euclidean and well complemented.

Challenged assumptions about the structure of Euclidean subspaces in normed spaces.

Abstract

Let $|\cdot|$ be the standard Euclidean norm on $\mathbb{R}^n$ and let $X=(\mathbb{R}^n,\|\cdot\|)$ be a normed space. A subspace $Y\subset X$ is \emph{strongly $\alpha$-Euclidean} if there is a constant $t$ such that $t|y|\leq\|y\|\leq\alpha t|y|$ for every $y\in Y$, and say that it is \emph{strongly $\alpha$-complemented} if $\|P_Y\|\leq\alpha$, where $P_Y$ is the orthogonal projection from $X$ to $Y$ and $\|P_Y\|$ denotes the operator norm of $P_Y$ with respect to the norm on $X$. We give an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is both strongly $(1+\epsilon)$-Euclidean and strongly $(1+\epsilon)$-complemented, where $\epsilon>0$ is an absolute constant. This example is closely related to an old question of Vitali Milman.