Carathéodoryʹs Theorem and H-Convexity

In 1976, V. Boltyanski introduced the functional md for compact, convex bodies. With the help of this functional, some theorems of combinatorial geometry were derived. For example, the first author obtained a Helly-type theorem, later some particular cases of the Szökefalvi–Nagy problem were resolved. Further on, exact estimates for the cardinalities of primitive fixing and hindering systems of compact, convex bodies were established, etc. In this article, we discuss the connection of the classical Carathéodory Theorem with the functional md.