One introduces a conjugacy relation and a subdifferential on the class of functions \(g:R^ n\to\bar R.\) It is shown that \(g^{**}=g\) iff g is proper (i.e. \(g(0)=\sup g),\) homogeneous of degree zero and evenly quasi- convex. As noted by the authors, the above notions are much related to the ones introduced by \textit{J. P. Crouzeix} [Math. Oper. Res. 5, 120-125 (1980; Zbl 0428.26007)] and \textit{H. J. Greenberg} and \textit{W. P. Pierskalla} [J. Optimization Theory Appl. 16, 409-428 (1975; Zbl 0302.90047)], respectively, through the extension of g: \(f_ g:R^{n+1}\to\bar R\), \(f_ g(x,\lambda)=g(x/\lambda)\) if \(\lambda >0,\quad f_ g(x,\lambda)=\sup g\) if \(\lambda\leq 0.\) Properties of the conjugate and the subdifferential are stated and the relations between the conjugate and the Legendre transform.