As many antipodes as vertices on convex polyhedra

J. Rouyer (1) , Tewfik Sari (2, 3)

1 LMIA - Laboratoire de Mathématiques Informatique et Applications [UHA]
2 UMR ITAP - Information – Technologies – Analyse Environnementale – Procédés Agricoles
3 MODEMIC - Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme

An earlier result states that, on the surface of a convex polyhedron with n vertices endowed with its intrinsic metric, a point cannot have more than n antipodes (farthest points). In this paper we produce examples of polyhedra with n vertices, on which some suitable point admits exactly n antipodes. We also proved that, for any positive number d ≤ 1, there exist (in the closure of the set of these polyhedra) some convex surfaces on which some point have a set of antipodes of Hausdorff dimension d.

Mots clés: GEOMETRIE; FORMULE MATHEMATIQUE

Domaines: Sciences de l'environnement

J. Rouyer, Tewfik Sari. As many antipodes as vertices on convex polyhedra. Advances in Geometry, 2012, 12 (1), pp.43 -61. ⟨10.1515/advgeom.2011.030⟩. ⟨hal-00777554⟩