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On the form of the unit ball of the one-dimensional stable norm. (Sur la forme de la boule unité de la norme stable unidimensionnelle.) (French) Zbl 1082.05509

Summary: We study the stable norm on the first homology of a Riemannian polyhedron. In the one-dimensional case (metric graphs), the geometry of the unit ball of this norm is completely described by the combinatorial structure of the graph. For a smooth manifold of dimension \(\geq 3\) and using polyhedral techniques, we show that a large class of polytopes appears as unit ball of the stable norm associated to some metric conformal to a given one.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
05C38 Paths and cycles
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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