On the cone of a finite dimensional compact convex set at a point (Q1820434)

This paper centers around the following problem formulated by the second author (1983): Is it true that given any cone K in V (a Euclidean space) there always exists a compact convex set C whose point's cone at some given point y is equal to K ? The authors answer this question negatively by establishing their main result: On a cone K in V the following statements are equivalent: 1) There exists a compact convex set C containing 0 such that \(cone (0,C)=K.\) (Here \(cone (y,C)\) denotes the cone \(\{\) t(x-y)\(| t\geq 0\) and \(x\in C\}\) called a point's cone). 2) K is an \(F_{\sigma}\)-set. 3) There exists a nondecreasing sequence \((K_ i)_{i\in N}\) of closed cones whose union is K. 4) K is the barrier cone of some nonempty convex set. (The barrier cone of the convex set C is \(B(C)=\{y\in V| \sup_{x\in C}(x,y)<\infty \}\), where (, ) denotes the inner product in V.) The authors also relate their result to the point classification of finite dimensional convex sets by \textit{Z. Waksman} and \textit{M. Epelman} [Math. Scand. 38, 83-96 (1976; Zbl 0331.52004)].