Summary: Prox-regularity of a set [{\it R. A. Poliquin, R. T. Rockafellar} and {\it L. Thibault}, Trans. Am. Math. Soc. 352, No. 11, 5231--5249 (2000;

Zbl 0960.49018)], or its global version, proximal smoothness [{\it F. H. Clarke, R. J. Stern} and {\it P. R. Wolenski}, J. Convex Anal. 2, No. 1--2, 117--144 (1995;

Zbl 0881.49008)] plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function $\text{dist}\,(C;\cdot)$, or the local uniqueness of the projection mapping, but also because in the case where $C$ is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-$C^{2}$ property) of the function. In this paper we provide an adapted geometrical concept, called subsmoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-$C^{1}$ property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by {\it A. Lewis} [“Robust regularization”, preprint (2002)]. We hereby relate it to the Mifflin semismooth functions.