Analytic parametrization and volume minimization of three dimensional bodies of constant width

Térence Bayen (1) , Thomas Lachand-Robert (2) , Edouard Oudet (2, 3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 LAMA - Laboratoire de Mathématiques
3 Département de mathématiques

We present a complete analytic parametrization of constant width bodies in dimension 3 based on the median surface: more precisely, we define a bijection between some space of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke's formula. Finally, we derive weak optimality conditions for convex bodies which minimize the volume among constant width bodies.

Domaines: Equations aux dérivées partielles [math.AP]

Térence Bayen, Thomas Lachand-Robert, Edouard Oudet. Analytic parametrization and volume minimization of three dimensional bodies of constant width. Archive for Rational Mechanics and Analysis, 2007, 186 (2), pp.225--249. ⟨hal-00385121⟩