  The "old-new" concept of convex-hull function was investigated by several
authors in the last seventy years. A recent research on it led to some other
volume functions as the covariogram function, the widthness function or the
so-called brightness functions, respectively. A very interesting fact that
there are many long-standing open problems connected with these functions whose
serious investigation closed before the "age of computers". In this survey, we
concentrate only on the three-dimensional case, we will mention the most
important concepts, statements, and problems.
