Geodesic
Some properties of Carnot-Carathéodory balls in the Heisenberg group
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 3, pp. 155-167.

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Using the exact representation of Carnot-Carathéodory balls in the Heisenberg group, we prove that: 1. $|\nabla_{\mathbb{H}^{n}} d(z,t)| = 1$ in the classical sense for all $(z,t) \in \mathbb{H}^{n}$ with $z \neq 0$, where $d$ is the distance from the origin; 2. Metric balls are not optimal isoperimetric sets in the Heisenberg group.
Usando la rappresentazione esatta per le sfere di Carnot-Carath´ eodory nel gruppo di Heisenberg, proviamo che: 1. $|\nabla_{\mathbb{H}^{n}} d(z,t)| = 1$ in senso classico per ogni $(z,t) \in \mathbb{H}^{n}$ con $z \neq 0$, dove $d$ è la distanza dall’origine; 2. Le sfere metriche non sono insiemi isoperimetrici ottimali nel gruppo di Heisenberg. 
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Monti, Roberto. Some properties of Carnot-Carathéodory balls in the Heisenberg group. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 3, pp. 155-167. http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_3_a1/

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