Intersection formulae for the kernel of a cone
A subset S of d-dimensional Euclidean space \({\mathbb{R}}^ d\) (d\(\geq 1)\) is called a cone if it is the union of a set of halflines having the same endpoint called the apex of the cone. The set of all such apices (each of which belongs to a representation of S as a cone) is called the \({\mathbb{R}}\)-kernel K(S) of S. In the present paper, K(S) is shown to be the intersection of a family of flats passing through selected boundary points of S. Three similar but independent formulae for K(S) are established, respectively: for an arbitrary proper subset S, for S closed, and for S closed connected and nonconvex. All of these formulae involve sets, convex and affine hulls of sets from which selected points of the boundary of S are ``R-visible'' (a point y of S is called R- visible from a point x of S, \(x\neq y\), via S iff the corresponding closed ray emanating from x through y lies in S) or even ``clearly R-visible'' via S. The three formulae may be useful for further studies on the dimension of the \({\mathbb{R}}\)-kernel of a cone S (which is a k-dimensional flat, \(0\leq k<d\), in \({\mathbb{R}}^ d\) if \(S\neq {\mathbb{R}}^ d)\).
cone in Euclidean space \({\mathbb{R}}^ d\)
scientific article; zbMATH DE number 4210950