Dictionary between Toland–Singer and local Legendre duality

Establish, under generic hypotheses, whether the Toland–Singer dual f⋄ and the local differential-geometric Legendre transform f* agree on an open dense subset of the dual domain D*.

Background

The paper distinguishes two dual constructions for a difference-of-convex potential: the local Legendre transform obtained from the inverse gradient map and the global Toland–Singer variational dual obtained from the difference of Fenchel conjugates.

Although the two notions coincide at points associated with global minimizers, the authors do not establish a general relationship between them. The unresolved issue is whether generic assumptions force agreement on a substantial, specifically open dense, portion of the dual domain.