Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

A uniqueness proof for the Wulff Theorem

Irene Fonsecaa1 and Stefan Müllera2

a1 Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.

a2 Institut für Angewandte Mathematik, Universität Bonn, Beringstr. 4, W-5300 Bonn 1, Germany


The Wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of (small) elastic crystals. It consists in minimising the (generally anisotropic) surface energy among sets of given volume. A solution of this problem is given by a geometric construction due to Wulff. In the class of sets of finite perimeter this was first shown by J. E. Taylor who, using methods of geometric measure theory, also proved uniqueness. Here a more analytic uniqueness proof is presented. The main ingredient is a sharpened version of the Brunn–Minkowski inequality.

(Received July 19 1990)

(Revised November 15 1990)