Comparing Fenchel-Moreau Conjugates with Level Set Conjugates

We compare the Fenchel-Moreau second conjugates associated to an arbitrary coupling function $\varphi:X\times W\rightarrow \overline{R}=[-\infty,+\infty ]$ between two sets $X$ and $W$ with the second level set conjugates associated to the same coupling. For a coupling $\varphi:R^{n}\times R^{n}\rightarrow R=(-\infty,+\infty )$ 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.