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

Fix $p\in[1,\infty)$, $K\in(0,\infty)$ and a probability measure $μ$. We prove that for every $n\in\mathbb{N}$, $\varepsilon\in(0,1)$ and $x_1,\ldots,x_n\in L_p(μ)$ with $\big\| \max_{i\in\{1,\ldots,n\}} |x_i| \big\|_{L_p(μ)} \leq K$, 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 $\forall \ i,j\in\{1,\ldots,n\}, \qquad \|x_i-x_j\|^p_{L_p(μ)}- \varepsilon \leq \|y_i-y_j\|_{\ell_p^d}^p \leq \|x_i-x_j\|^p_{L_p(μ)}+\varepsilon.$ 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 $L_p(μ)$ 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 $p\neq2$.

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