Convex Along Lines Functions and Abstract Convexity. Part I

\def\RR{\mathbb R} \def\rc#1{{\rm #1\,}} The present paper investigates the property of a function $f\colon \RR^n \to \RR_{+\infty}:= \RR \cup \{+\infty\}$ with $f(0) < +\infty$ to be ${\cal L}_n$-subdifferentiable or $\calH_n$-convex. The $\calL_n$-subdifferentiability and $\calH_n$-convexity are introduced as in the book of A. M. Rubinov [``Abstract convexity and global optimization'', Kluwer Academic Publishers, Dordrecht 2000]. Some refinements of these properties lead to the notions of $\calL_n^0$-subdifferentiability and $\calH_n^0$-convexity. Their relation to the convex-along (CAL) functions is underlined in the following theorem proved in the paper (Theorem 5.2): Let the function $f\colon \RR^n \to \RR_{+\infty}$ be such that $f(0) < +\infty$ and $f$ is $\calH_n$-convex at the points at which it is infinite. Then if $f$ is $\calL_n^0$-subdifferentiable, it is CAL and globally calm at each $x^0\in\rc{dom}f$. Here the notions of local and global calmness are introduced after R. T. Rockafellar and R. J-B Wets [``Variational analysis'', Springer-Verlag, Berlin 1998] and play an important role in the considerations. The question is posed for the possible reversal of this result. In the case of a positively homogeneous (PH) and CAL function such a reversal is proved (Theorems 6.2). As an application conditions are obtained under which a CAL PH function is $\calH_n^0$-convex (Theorems 6.3and 6.4).