Frank Morgan

The following old post-publication correspondence provides some information and questions on the original proof that the Wulff shape minimizes anisotropic surface energy for fixed volume, as treated in our paper:

MR1297699 (95g:49080)
Brothers, John E. (1-IN); Morgan, Frank (1-WLMS)
The isoperimetric theorem for general integrands.
Michigan Math. J. 41 (1994), no. 3, 419–431.
49Q20

From: [email protected]
Subject: Gromov reference
Date: September 3, 1994 at 3:50:28 PM EDT
To: Brothers
Cc: Hutchings, Taylor

Dear John,

After all this time I have found the mythical original Gromov proof of the isoperimetric inequality:

[G] M. Gromov, Isoperimetric inequalities in Riemannian manifolds, Appendix I to Vitali D. Milman and Gideon Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics, No. 1200, Springer-Verlag, 1986.

The big surprise to me was that Gromov treats general norms, which I thought was my idea. Indeed, he attributes the isoperimetric and Sobolev inequalities for general norms to Brunn’s inaugural dissertation in Munich in 1887, 14 years before Wulff! Through the NY Public Library I have learned that there are copies of Brunn’s dissertation at Brown, Michigan, and Princeton and I have requested it through interlibrary loan. I may send a further revision of this memo after I have seen it.

Gromov [G] points out uniqueness among smoothly bounded sets and remarks that the isoperimetric inequality for nonsmooth sets follows by approximation, but of course that does not prove uniqueness for the nonsmooth case, which really is the main accomplishment of our paper.

In his first remark on p. 127, I don’t know why he introduces the new hypothesis “convex” or why he omits the hypothesis “bounded.”

I would suggest the following revisions to our paper:

Abstract: put “uniquely” in italics

p. 2: after set-off definition of B_Phi: We have just seen a report by Gromov [G] that the isoperimetric inequality for general norms appeared earlier in Brunn’s inaugural dissertation [Bru].

p. 4: . . . has been used by Gromov {[G]; see [Be, 12.11.4] to prove the theorem, with uniqueness only for smoothly bounded sets. Gromov also points out the equivalence of Sobolev’s inequality. The mapping F was defined earlier by H. Knothe [K] for the purpose of deriving a generalized Brunn-Minkowski inequality. The application to proving uniqueness in the general case is new.

[Bru] H. Brunn, Ueber Ovale und Eiflaechen, Inaug. Diss., Muenchen, 1887.

[Ue stands for U umlaut, ae for a umlaut, ue for u umlaut.]

Note that here I use [Be] for Berger and [Bru] for Brunn, to distinguish them from the reference to you, which perhaps should be [Bro]. (I missed the potential Berger-Brothers confusion in my previous message of 9/9/93, copied below.)

I am getting settled in NYC for the fall, living at 210 W 15 St ((212) 989-8025) and teaching a course at Queens College.

Best,

Frank

Hi Frank–