Polygonal equalities in Hilbert spaces
  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. 
Anthony Weston
math.MG