From the authors 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 for \(\mathbb{H}^1\) to higher dimensional Heisenberg group (see [R. Monti and M. Rickly, J. Convex Anal. 12, No. 1, 187–196 (2005; Zbl 1077.53030)] where the notion of convexity for \(\mathbb{H}^1\) and, in particular, the term “geodetically convex” was introduced).