Summary: Acta Applicandae Mathematicae 57: 287338, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
On the Dido Problem and Plane
ANDREI A. AGRACHEV and JEAN-PAUL A. GAUTHIER
Laboratoire de Topologie, UMR 5584, University of Burgandy, BP 400, 21011, Dijon Cedex, France
(Received: 4 April 1999)
Abstract. This paper is a continuation of a series of papers, dealing with contact sub-Riemannian
metrics on R3. We study the special case of contact metrics that correspond to isoperimetric problems
on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at
least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It
appears that the case of generic isoperimetric problems falls down in the category of generic sub-
Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry).
Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude
immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem
of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details
this special case, for which we solve the problem generically (again, for generic cases, we compute
the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities).
In an addendum, we say a few words about: (1) the singularities that can appear in general for the