## Wulff Shape Isoperimetric

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

**From:**[email protected] (John Brothers)

**Subject:**

**Re: Gromov reference**

**Date:**September 9, 1994 at 2:37:00 PM EDT

**To:**<[email protected]>

Hi Frank–

Brunn. I began looking for the Gromov reference eight years ago. By

coincidence I returned the corrected galleys last week and so I think the

time is past for us to make revisions in the paper. They say it is

scheduled to come out by year’s end. Also, they declined to try to print

the illustrations of crystals, saying that they did not have the technology

to do so. A reference to the picture is included.

I hope you enjoy your year in New York. My son Max is now living in

Princeton, where he is working for a derivatives trading company.

John

**From:**[email protected]

**Subject:**

**Letter to Gromov on Brunn**

**Date:**September 24, 1994 at 12:41:50 PM EDT

**To:**Brothers

I got Brunn’s thesis of 1887, but I find nothing on general norms in it. It is in German in the old-fashioned style and hard for me to read. It probably has the bones of the Brunn-Minkowski Theorem. Since

Gromov actually said Brunn got general norms in 1888, perhaps they came shortly afterwards. I have written the letter below to Gromov to try to find out.

Best,

Frank

Chelsmore Apt. #5S

205 W 15 Street

New York, New York 10011

September 26, 1994

Prof. Mikhael Gromov

Institute des Hautes Etudes Sci.

91440 Bures-Sur-Yvette

France

Dear Mikhael,

I have just discovered your appendix on “Isoperimetric inequalities in Riemannian manifolds” to Milman and Schechtman’s Aymptotic Theory of Finite Dimensiona Normed Spaces (Lecture Notes in Mathematics 1200, 1986). I had thought I was the first to notice that the divergence theorem proof worked for general norms.

You attribute the theorem for general norms to Brunn in 1888. How do you know this? Is there any reference for this? The literature credits Wulff, 1901.

I would be grateful for any response by mail or better email to

[email protected]

Thanks and best wishes.

Sincerely,

Frank Morgan

Encl: related paper by Brothers and Morgan

cc: Brothers