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Geodetically convex sets in the Heisenberg group. (English) Zbl 1077.53030
Let \(H\) be the 3-dimensional Heisenberg group equipped with its usual left-invariant Carnot-Carathéodory metric. It is well-known that any two points in \(H\) can be joined by a geodesic. The main result of the paper is this: If three points in \(H\) do not lie on a common geodesic, then the only geodesically convex subset of \(H\) containing them is \(H\) itself. As a consequence, it follows that all geodesically convex functions on \(H\) are constant.

MSC:
53C17 Sub-Riemannian geometry
22E25 Nilpotent and solvable Lie groups
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