This document discusses convexity and H-convexity in Rn. It defines convexity as a collection of subsets that is stable under intersection and nested union. H-convexity is generated by half-spaces, which are subsets whose complements are convex. The document presents definitions of arity, which describes how convex sets are determined by subsets of bounded cardinality, and separation axioms S1-S4. It provides an example of a symmetric H-convexity generated by linear functionals on R2 and discusses a convexity of infinite arity on Rn generated by linear functionals.