$\varepsilon$-isometric dimension reduction for incompressible subsets of $\ell_p$

Abstract: Fix $p\in[1,\infty)$, $K\in(0,\infty)$ and a probability measure $\mu$. We prove that for every $n\in\mathbb{N}$, $\varepsilon\in(0,1)$ and $x_1,\ldots,x_n\in L_p(\mu)$ with $\big| \max_{i\in{1,\ldots,n}} |x_i| \big|{L_p(\mu)} \leq K$, there exists $d\leq \frac{32e^2 (2K)^{2p}\log n}{\varepsilon^2}$ and vectors $y_1,\ldots, y_n \in \ell_p^d$ such that $$\forall \ i,j\in{1,\ldots,n}, \qquad |x_i-x_j|^p{L_p(\mu)}- \varepsilon \leq |y_i-y_j|{\ell_p^d}^p \leq |x_i-x_j|^p{L_p(\mu)}+\varepsilon.$$ Moreover, the argument implies the existence of a greedy algorithm which outputs ${y_i}{i=1}^n$ after receiving ${x_i}{i=1}^n$ as input. The proof relies on a derandomized version of Maurey's empirical method (1981) combined with a combinatorial idea of Ball (1990) and classical factorization theory of $L_p(\mu)$ spaces. Motivated by the above embedding, we introduce the notion of $\varepsilon$-isometric dimension reduction of the unit ball ${\bf B}E$ of a normed space $(E,|\cdot|_E)$ and we prove that ${\bf B}{\ell_p}$ does not admit $\varepsilon$-isometric dimension reduction by linear operators for any value of $p\neq2$.