Quantitative stability estimates for the Brunn-Minkowski inequality
1. The stability problem
1.1. Brunn-Minkowski’s inequality
Let . Note that , so
When does equality hold ? When . This is an easy particular case of the following.
Theorem 1 (Brunn-Minkowski) Let and be sets of positive volume in . Then
To get convinced why exponents need be there, take balls.
When does equality hold ? When , are convex and homothetic, up to sets of measure 0.
Brunn-Minkowski’s inequality is related, via isoperimetric and Sobolev inequalities, to all functional inequalities.
1.2. Are minimizer stable ?
For instance, for Sobolev inequality
imply that is close to a minimizer ?
We are concerned with a similar question for Brunn-Minkowski’s inequality.
1.3. Quantitative formulation
Let and . Set
Is it true that, as tends to 0, and become nearly convex and nearly homothetic ? Of course, we want something quantitative. We separate the convexity issue from the homothety issue.
So we assume first that and are convex. Set
so that . Set
Question. Find an estimate of in terms of .
2. The homothety issue
2.1. A result dating back to 2009
Theorem 2 (Figalli-Maggi-Pratelli 2009) There exists a constant such that
Why a square root ? behaves as if it were a smooth function on the space of pairs of convex sets of equal volumes, with a nondegenerate minimum at homothetic pairs.
2.2. Proof of Brunn-Minkowski’s inequality
Based on optimal transport. Consider normalized Lebesgue measures on and . Let be the optimal transport map. Optimal transport theory says that exists, it has constant Jacobian on and it is the gradient of a convex function on .
contains all points , so
Diagonalize . Then
It follows that
3. The convexity issue
3.1. One dimensional case
Assume first that . Does control ? No.
Example 1 Let be the union of two remote intervals of length in .
Then contains a third interval of length in the middle, so , .
It turns out that the 1-dimensional case has been known for a while. It follows from Freiman’s theorem in additive combinatorics.
Theorem 3 (Freiman 1959) Let have and . Then
In fact, Freiman’s theorem is much stronger. It deals with subsets in , and shows that close to implies that is close to an arithmetic progression.
3.2. Higher dimensions
Theorem 4 (M. Christ, 2012) tends to 0 as tends to 0.
Theorem 5 (Figalli-Jerison, 2013) If ,
This uses induction on dimension, based on the 1-dimensional result.
3.3. Sketch of proof
Slice with parallel lines , . Show that is close to a segment for most . Furthermore, the projection of to is close to be convex.
is close to an affine function.
Let be a Lipschitz function on a set . Assume that
where . Then is close to a concave function ,
3.4. General case
In 1 dimension, Freiman’s theorem does the job. Michael Christ’s qualitative result also extends.
Theorem 6 (Figalli-Jerison, 2014) If , there exists a convex set such that
Let . Since
Fubini seems to show that
which was the first step of our proof when . But this argument collapses when , since it happens that is empty and is not, in which case Brunn-Minkowski’s inequality fails!
To avoid this, we perform two preliminary symmetrizations, Schwarz and Steiner, in orthogonal directions. This preserves and also volumes of superlevel sets of .