Some characterizations of a uniform ball property*

In 1973, Helfrich suggested a simple model to describe the shapes of vesicles: a free bending energy involving geometric quantities like curvature. However, the mathematical questions concerning the existence and the regularity of minimizers to such shape optimization problems still remain open. In this article, we consider a class of admissible shapes in which the existence of minimizers is ensured: the hypersurfaces of Rn satisfying a uniform ball condition. We prove that this property is equivalent to the notion of positive reach introduced by Federer in 1959. Then, another characterization in terms of C1,1-regularity is established for compact hypersurfaces