Nondegenerate families of convex cones and convex polytopes

This paper describes relations between convex polytopes and certain families of convex cones in R ⁿ .The purpose is to use known properties of convex cones in order to solve Helly type problems for convex sets in R ⁿ or for spherically convex sets in S _n , the n-dimensional unit sphere. These results are strongly related to Gale diagrams.