Properties and applications of DC functions

A function in ${\mathbb{R}}^n$ is called DC (d.c., delta-convex) if it is the difference of two continous functions $f_1$, $f_2$ (i.e., $f=f_1-f_2$). DC functions are a natural and important joint generalization of both $C^2$ smooth and convex functions. I will recall some properties of DC functions and will give some information on several (older and quite recent) their applications: a)\ to singularities of convex functions; b)\ to ``distance spheres'' in Riemann spaces and in convex surfaces; c)\ to the theory of curvature measures of non-regular sets. Several open questions will be presented.