Ravines of functions and nonuniformity of their supergraphs

We introduce the notion of an $L$-ravine of a function, which is a generalization of the notion of a $c$-ravine introduced in [1], and give examples of functions, including convex polynomials, with different structures of $L$-ravines.
A connection of this notion with non-uniformity of the distribution of integer points, or generally of lattice nodes, in epigraphs of functions is demonstrated. In particular, it is proved that there exist absolutely non-uniform convex polynomials and convex functions in two variables which have no $c$-ravines but have $L$-ravines.