Krasnosel'skii-type characterizations for a cone
Generalizing Krasnosel'skii's theorem on starshaped sets the author proves the following characterizations of cones in Euclidean \(d\)-space \(\mathbb{R}^d\): 1. A closed set \(S\) in \(\mathbb{R}^d\) is a cone iff every \(d+1\) spherical points of \(S\) are \(R\)-visible from some point of \(S\). 2. If \(S\) is, in addition, convex then \(S\) is a cone iff any two points of the boundary of \(S\) are \(R\)-visible from some point of \(S\). 3. A closed connected non-convex set in \(\mathbb{R}^3\) is a cone iff two locally non-convex points of \(S\) are clearly \(R\)-visible via \(S\) from some point of \(S\). Here, \(y\in S\) is called \(R\)-visible from \(x\in S\) if the entire ray from \(x\) through \(y\) belongs to \(S\). \(y\in S\) is called clearly \(R\)-visible from \(x\in S\) if some relative neighborhood of \(y\) in \(S\) is \(R\)-visible from \(x\). Being spherical and locally non-convex are support-properties of boundary points of the set \(S\). The proofs are of a generic nature [confer e.g. \textit{F. A. Valentine}, Convex sets. New York etc.: McGraw-Hill (1964; Zbl 0129.37203)] though involved, especially that of the 3rd statement.
scientific article; zbMATH DE number 12142
scientific article