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

A geodetically convex set in the Heisenberg group \({\mathbb {H}}^n\) ${\mathbb{H}}^n$ , \(n\ge 1\) $n\ge 1$ 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 \({\mathbb {H}}^n\) ${\mathbb{H}}^n$ . We also show that a geodetically convex function on \({\mathbb {H}}^n\) ${\mathbb{H}}^n$ is a constant function. These results generalize the known results of \({\mathbb {H}}^1\) ${\mathbb{H}}^1$ to higher dimensional Heisenberg group.