On closed weakly m-convexsets

Abstract

In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n>1), known as weakly m-convex, m=1,...,n-1. An open set of Rn is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed set of Rn is called weakly m-convex if it is approximated from the outside by a family of open weakly m-convex sets. A point of the complement of a set of Rn to the whole space is called an m-nonconvexity point of the set if any m-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly (n-1)-convex set in Rn with non-empty set of (n-1)-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly 1-convex set with a finite number of components in the plane is weakly 1-convex. Weakly m-convex domains and closed connected sets in Rn with non-empty set of m-nonconvexity points are constructed for any n>2 and any m=1,...,n-1.