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.

中文翻译：

---

海森堡群中的大地凸集 $${\mathbb {H}}^n$$ , $$n \ge 1$$

海森堡群中的大地测量凸集\({\mathbb {H}}^n\) , \(n\ge 1\)被定义为具有以下属性的集合：连接该集合中任意两点的测地线位于完全在其中。在这里，我们将大地凸集分类为空集、单例集、测地线弧或整个空间\({\mathbb {H}}^n\)。我们还证明了\({\mathbb {H}}^n\)上的大地测量凸函数是常数函数。这些结果将\({\mathbb {H}}^1\)的已知结果推广到更高维的海森堡群。