Sur la forme de la boule unité de la norme stable unidimensionnelle

Ivan K. Babenko (1) , Florent N. Balacheff (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then the realizable forms as stable ball of Riemannan manifolds of dimension larger than three. For a Riemannian manifold $(M, g)$ fixed, we show that a broad class of polytopes can appear as stable ball of metrics in the conformal class of $g$. We use for that a polyhedral technique.

Mots clés: Riemannian manifold; Riemannian polyhedra; stable norm; unit ball

Domaines: Géométrie différentielle [math.DG]; Combinatoire [math.CO]; Géométrie métrique [math.MG]

Ivan K. Babenko, Florent N. Balacheff. Sur la forme de la boule unité de la norme stable unidimensionnelle. Manuscripta mathematica, 2006, 119 (3), pp.347-358. ⟨10.1007/s00229-005-0622-x⟩. ⟨hal-00004322⟩