Caffarelli Milman

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The reference for the paper discussed is

Caffarelli, Luis A.; Milman, Mario. The regularity of planar maps with bounded compression and rotation. Volume in homage to Dr. Rodolfo A. Ricabarra (Spanish), 29--34, Vol. Homenaje, 1, Univ. Nac. del Sur, Bahía Blanca, 1995.

The paper is not online, and the volume is not in our collection here at U of T. A pdf containing a scan can be downloaded here.

A related paper which treats some issues more extensively, including measurability properties of monotone mappings and affinely invariant notions of rotation, is

Caffarelli, Luis A. The regularity of monotone maps of finite compression. Comm. Pure Appl. Math. 50 (1997), no. 6, 563--591.

This paper does not restrict to the two-dimensional case, and proves a Hölder regularity result. In the appendix one finds two interesting counterexamples which were alluded to in the first part of the talk.

Caffarelli has written an interesting note on regularity and counterexamples for convex solutions of the Monge-Ampère equation, in which he shows that a certain type of degeneracy can only occur on a set of dimension $< n / 2$ (hence the special nature of the $n = 2$ case).

Caffarelli, Luis A. A note on the degeneracy of convex solutions to Monge Ampère equation. Comm. Partial Differential Equations 18 (1993), no. 7-8, 1213--1217.

In the talk I needed an inequality; there was a moment of confusion surrounding its derivation. The fact was that given a $2\times 2$ matrix $D$ with symmetric part $S = (D + D t) / 2$ , we have

$2\sqrt{\det S} = \inf_{t>0, T\in SO(2)} t (T^tDT)_{11} + \frac{1}{t} (T^tDT)_{22}$ .

Note that $T t D T$ has symmetric part $T t S T$ , so these two matrices have the same diagonal entries. The infimum is attained when $T$ diagonalizes $S$ and $t$ is such that the two terms become equal. To prove it, first take the inf over t>0 for a given T, obtaining $2\sqrt{(T^tST)_{11}(T^tST)_{22}}$ . As $T$ ranges over $S O (2)$ , $det(T t S T)$ stays constant; the diagonal term of the determinant is minimized when the other term, which contributes nonpositively, becomes zero, i.e. when $T t S T$ is diagonal. (I believe the confusion related to the eigenvalues of $D$ , which are not really involved here -- $D$ is not getting diagonalized at all.)

While the above was an ad-hoc technical device for the development, it is essentially a special case of the following nice fact, which stems from the arithmetic-geometric mean inequality.

Let $S_n^+(\mathbb{R})$ denote the convex cone of positive symmetric $n\times n$ matrices. The group $SL_n(\mathbb{R})$ of matrices of unit determinant acts on this cone by affine conjugation, i.e. $S\in S_n^+, T\in SL_n \Rightarrow T^tST\in S_n^+$ . Note that the function ${\det}^{1/n} : S_n^+ \rightarrow \mathbb{R}^+$ is invariant under this action; in fact, the orbits of the action are precisely the level sets of $det 1 / n$ . Indeed, given $S$ , take $T = O P$ , with $O$ an orthogonal matrix diagonalizing $S$ , and $P$ the positive diagonal matrix of unit determinant whose eigenvalues are inversely proportional to the square roots of the respective eigenvalues of $S$ ; we then have that $T t S T = det 1 / n (S) I$ is scalar.

Now we observe that

${\det}^{1/n}(S) = \textstyle\frac{1}{n}\displaystyle \inf_{T\in SL_n} \mathrm{tr}\,(T^t S T)$ .

This is a direct consequence of the arithmetic-geometric mean inequality, applied to the eigenvalues of $T t S T$ . Equality holds for the above $T$ that brings $S$ to a scalar. One can view this as a representation of the concave function $det 1 / n$ as the lower envelope of a family of linear functions. (N.B. linear functions, not merely affine functions... $det 1 / n$ is flat along the rays of the cone $S_n^+$ , though it is strictly concave along all other segments (I think).)

Caffarelli Milman

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