Several theorems of combinatorial geometry

A set is said to be $H$-convex if it can be represented by an intersection of a family of closed half-spaces whose outer normals belong to a given subset of the set $H$ of the unit sphere $S^{n-1}\subset R$. On the basis of Helly's theorem for $H$-convex sets recently obtained by us, we prove in this note certain extensions of Blaschke's theorem (on the radius of an inscribed sphere) and of several other well-known theorems of combinatorial geometry.