** 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

does

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 1Let 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 **

\subsubsection{Step 1}

Slice with parallel lines , . Show that is close to a segment for most . Furthermore, the projection of to is close to be convex.

\subsubsection{Step 2}

is close to an affine function.

\subsubsection{Step 3}

Let be a Lipschitz function on a set . Assume that

where . Then is close to a concave function ,

** 3.4. General case **

I.e. .

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 .