The geometry of two-valued subsets of $L_{p}$-spaces

TL;DR

This paper characterizes two-valued subsets of $L_p$ spaces that have strict negative type, linking geometric independence to metric properties, and constructs examples with specific negative type characteristics.

Contribution

It establishes a precise criterion for two-valued subsets to have strict $p$-negative type based on affine independence, and introduces methods to construct subsets with tailored negative type properties.

Findings

Two-valued subsets with affine independence have strict $p$-negative type.

Every two-valued Schauder basis in $L_p$ has strict $p$-negative type.

Constructed subsets with $p$-negative type but not $q$-negative type for $q > p$.

Abstract

Let $\mathcal{M}(\Omega, \mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(\Omega, \mu)$. Let $B \subset \mathcal{M}(\Omega, \mu)$ be a set of finitely supported measurable functions such that the essential range of each $f \in B$ is a subset of $\{ 0,1 \}$. The main result of this paper shows that for any $p \in (0, \infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(\Omega, \mu)$ if and only if $B$ is an affinely independent subset of $\mathcal{M}(\Omega, \mu)$ (when $\mathcal{M}(\Omega, \mu)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_{p}(\Omega, \mu)$ has strict $p$-negative type. For instance, for each $p \in (0, \infty)$, the system of Walsh functions in $L_{p}[0,1]$ is seen to have strict $p$-negative type. The techniques developed in this paper also provide a systematic way to construct, for any $p \in (2, \infty)$, subsets of $L_{p}(\Omega, \mu)$ that have $p$-negative type but not $q$-negative type for any $q > p$. Such sets preclude the existence of certain types of isometry into $L_{p}$-spaces.