Comparing Fenchel-Moreau Conjugates with Level Set Conjugates

We compare the Fenchel-Moreau second conjugates associated to an arbitrary coupling function

between two sets

X X

and

W W

with the second level set conjugates associated to the same coupling. For a coupling

\varphi:R^{n}\times R^{n}\rightarrow R=(-\infty,+\infty ) φ : R n × R n → R = ( − ∞ , + ∞ )

that is additively homogeneous in one (or both) of the variables we also compare the first conjugates associated to the same coupling. We give an application to the “min-type” coupling function arising in the study of topical functions.