Polygonal equalities in Hilbert spaces

TL;DR

This paper characterizes equality cases in 2-negative type inequalities in Hilbert spaces, linking strict negative type to affine independence and describing the class M of metric subspaces.

Contribution

It provides a complete characterization of equality in 2-negative type inequalities and relates strict negative type to affine independence in Hilbert spaces.

Findings

Equality cases characterized by balanced signed simplices

Strict 2-negative type iff the subspace is affinely independent

Complete description of Shkarin's class M

Abstract

This work has been expanded and fully incorporated into arXiv:1203.5837. Cases of equality in the classical 2-negative type inequalities for Hilbert spaces are characterized in terms of balanced signed simplices. It follows that a metric subspace of a Hilbert space H has strict 2-negative type if and only if it is affinely independent (when H is considered as a real vector space). This allows a complete description of Shkarin's class M.