Caratheodory's theorem and H-convexity

In 1976, V. Boltyanski introduced the functional md for compact, convex bod ies. With the help of this functional, some theorems of combinatorial geome try were derived. For example. the first author obtained a Helly-type theor em, later some particular cases of the Szokefalvi-Nagy problem were resolve d. Further on, exact estimates for the cardinalities of primitive fixing an d hindering systems of compact, convex bodies were established. etc. In thi s article. we discuss the connection of the classical Caratheodory Theorem with the functional md. (C) 2001 Academic Press.