Abstract Convexity and Connectedness

In a former paper the concept of n-ary connectedness was introduced, where 1-ary connectedness coincides with the usual notion of (abstract) connectedness. In the present paper, sets endowed with a convexity structure are studied, where the polytopes are n-ary connected. Interrelations between the classical Helly and Carathéodory numbers are evaluated as a pre-stage of Helly and Carathéodory type intersection theorems. Various other applications such as intersection theorems and fixed point theorems for trees and hyperconvex metric spaces are presented.