Proof
Various proofs of the theorem have been given by Hilton, Liebman, von Laue, Herring and a rather extensive treatment by Cerf. The following is after the method of R. F. Strickland-Constable. We begin with the surface energy for a crystal
which is the product of the surface energy per unit area times the area of each face, summed over all faces, which is minimized for a given volume when
We then consider a small change in shape for a constant volume
which can be written as
the second term of which must be zero, as it represents the change in volume, and we wish only to find the lowest surface energy at a constant volume (i.e. without adding or removing material.) We are then given from above
and
which can be combined by a constant of proportionality as
The change in shape must be allowed to be arbitrary, which then requires that which then proves Gibbs-Wulff Theorem.
Read more about this topic: Wulff Construction
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