A doubling subset of $L_p$ for $p>2$ that is inherently infinite dimensional

TL;DR

The paper constructs a doubling subset within Lp spaces for p>2 that cannot be embedded into any finite-dimensional Euclidean space via bi-Lipschitz maps, highlighting intrinsic infinite-dimensional complexity.

Contribution

It introduces a specific doubling subset of Lp for p>2 that inherently resists finite-dimensional Euclidean embedding, revealing new geometric properties of Lp spaces.

Findings

Existence of a doubling subset in Lp for p>2 that is not bi-Lipschitz embeddable into any finite-dimensional Euclidean space.

Demonstrates intrinsic infinite-dimensionality of certain subsets of Lp spaces.

Highlights limitations of finite-dimensional approximations of Lp geometries.

Abstract

It is shown that for every $p\in (2,\infty)$ there exists a doubling subset of $L_p$ that does not admit a bi-Lipschitz embedding into $\R^k$ for any $k\in \N$.