Geodetically convex sets in the Heisenberg group $${\mathbb {H}}^n$$ , $$n \ge 1$$

Abstract

A geodetically convex set in the Heisenberg group \({\mathbb {H}}^n\), \(n\ge 1\) is defined to be a set with the property that a geodesic joining any two points in the set lies completely in it. Here we classify the geodetically convex sets to be either an empty set, a singleton set, an arc of a geodesic or the whole space \({\mathbb {H}}^n\). We also show that a geodetically convex function on \({\mathbb {H}}^n\)is a constant function. These results generalize the known results of \({\mathbb {H}}^1\) to higher dimensional Heisenberg group.