## Notes of Alessio Figalli’s lecture

Quantitative stability estimates for the Brunn-Minkowski inequality

1. The stability problem

1.1. Brunn-Minkowski’s inequality

Let ${A\subset {\mathbb R}^n}$. Note that ${A\subset\frac{A+A}{2}}$, so

$\displaystyle \begin{array}{rcl} |A|\leq|\frac{A+A}{2}|. \end{array}$

When does equality hold ? When ${|Conv(A)\setminus A|=0}$. This is an easy particular case of the following.

Theorem 1 (Brunn-Minkowski) Let ${A}$ and ${B}$ be sets of positive volume in ${{\mathbb R}^n}$. Then

$\displaystyle \begin{array}{rcl} |\frac{A+B}{2}|^{1/n}\geq \frac{|A|^{1/n}+|B|^{1/n}}{2}. \end{array}$

To get convinced why exponents ${1/n}$ need be there, take balls.

When does equality hold ? When ${A}$, ${B}$ 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

$\displaystyle \begin{array}{rcl} \|u\|_q \leq C\|\nabla u\|_p, \end{array}$

does

$\displaystyle \begin{array}{rcl} \leq C\|\nabla u\|_p-\|u\|_q \ll 1 \end{array}$

imply that ${u}$ is close to a minimizer ?

We are concerned with a similar question for Brunn-Minkowski’s inequality.

1.3. Quantitative formulation

Let ${|A|\geq\lambda}$ and ${|B|\leq\Lambda}$. Set

$\displaystyle \begin{array}{rcl} \delta=|\frac{A+B}{2}|^{1/n}- \frac{|A|^{1/n}+|B|^{1/n}}{2}. \end{array}$

Is it true that, as ${\delta}$ tends to 0, ${A}$ and ${B}$ 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 ${A}$ and ${B}$ are convex. Set

$\displaystyle \begin{array}{rcl} \gamma=\frac{|A|^{1/n}}{|B|^{1/n}}, \end{array}$

so that ${|\gamma B|=|A|}$. Set

$\displaystyle \begin{array}{rcl} d(A,B)=\inf_{x\in{\mathbb R}^n}|A\Delta(\gamma B+x)|. \end{array}$

Question. Find an estimate of ${d(A,B)}$ in terms of ${\delta}$.

2. The homothety issue

2.1. A result dating back to 2009

Theorem 2 (Figalli-Maggi-Pratelli 2009) There exists a constant ${C=C(n,\lambda,\Lambda)}$ such that

$\displaystyle \begin{array}{rcl} d(A,B)\leq C\sqrt{\delta}. \end{array}$

Why a square root ? ${\delta}$ 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 ${A}$ and ${B}$. Let ${T:{\mathbb R}^n\rightarrow{\mathbb R}^n}$ be the optimal transport map. Optimal transport theory says that ${T}$ exists, it has constant Jacobian on ${A}$ and it is the gradient of a convex function ${\phi}$ on ${{\mathbb R}^n}$.

${A+B}$ contains all points ${a+Ta}$, so

$\displaystyle \begin{array}{rcl} |A+B|\geq|(Id+T)(A)|=\int_{A}\mathrm{det}(Id+\nabla\phi). \end{array}$

Diagonalize ${\nabla\phi}$. Then

$\displaystyle \begin{array}{rcl} \mathrm{det}(Id+\nabla\phi)&=&\prod_{i=1}^n(1+\lambda_i)\\ &\geq&(1+(\prod_{i=1}^n \lambda_i)^{1/n})^n\\ &=&(1+\gamma^{-1})^n. \end{array}$

It follows that

$\displaystyle \begin{array}{rcl} |A+B|\geq|A|(1+\gamma^{-1})^n=(|A|^{1/n}+|B|^{1/n})^n. \end{array}$

3. The convexity issue

3.1. One dimensional case

Assume first that ${A=B}$. Does ${\delta}$ control ${|Conv(A)\setminus A|}$ ? No.

Example 1 Let ${A}$ be the union of two remote intervals of length ${1/2}$ in ${{\mathbb R}}$.

Then ${\frac{A+A}{2}}$ contains a third interval of length ${1/2}$ in the middle, so ${|\frac{A+A}{2}|=\frac{3}{2}}$, ${\delta(A)=\frac{1}{2}}$.

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 ${A\subset {\mathbb R}}$ have ${|A|=1}$ and ${\delta(A)\leq\frac{1}{2}}$. Then

$\displaystyle \begin{array}{rcl} |Conv(A)\setminus A|\leq 2\delta(A). \end{array}$

In fact, Freiman’s theorem is much stronger. It deals with subsets in ${{\mathbb Z}}$, and shows that ${|A+A|}$ close to ${|A|}$ implies that ${A}$ is close to an arithmetic progression.

3.2. Higher dimensions

Theorem 4 (M. Christ, 2012) ${|Conv(A)\setminus A|}$ tends to 0 as ${\delta(A)}$ tends to 0.

Theorem 5 (Figalli-Jerison, 2013) If ${\delta(A)\leq\delta_n}$,

$\displaystyle \begin{array}{rcl} |Conv(A)\setminus A|\leq C_n\delta(A)^{\alpha_n}. \end{array}$

This uses induction on dimension, based on the 1-dimensional result.

3.3. Sketch of proof

\subsubsection{Step 1}

Slice ${A}$ with parallel lines ${D_y}$, ${y\in{\mathbb R}^{n-1}}$. Show that ${A_y}$ is close to a segment for most ${y}$. Furthermore, the projection of ${A}$ to ${{\mathbb R}^{n-1}}$ is close to be convex.

\subsubsection{Step 2}

${y\rightarrow|A_y|}$ is close to an affine function.

\subsubsection{Step 3}

Let ${f:K\rightarrow{\mathbb R}}$ be a Lipschitz function on a set ${K}$. Assume that

$\displaystyle \begin{array}{rcl} f(\frac{x+y}{2})\leq\frac{f(x)+f(y)}{2}+\gamma, \end{array}$

where ${\gamma=|Conv(A)\setminus A|}$. Then ${f}$ is ${L^1}$ close to a concave function ${F}$,

$\displaystyle \begin{array}{rcl} \int_{K}|f-F|\leq C\,\gamma^\alpha. \end{array}$

3.4. General case

I.e. ${B\not=A}$.

In 1 dimension, Freiman’s theorem does the job. Michael Christ’s qualitative result also extends.

Theorem 6 (Figalli-Jerison, 2014) If ${\delta(A,B)\leq\delta_n}$, there exists a convex set ${K}$ such that

$\displaystyle \begin{array}{rcl} |K\Delta A|+|K\Delta B|\leq C_n\delta(A,B)^{\beta_n}. \end{array}$

Let ${|A|=|B|=1}$. Since

$\displaystyle \begin{array}{rcl} \frac{A_y+B_y}{2}\subset (\frac{A+B}{2})_y, \end{array}$

Fubini seems to show that

$\displaystyle \begin{array}{rcl} |\frac{A+B}{2}|-\frac{|A|+|B|}{2}\geq 0, \end{array}$

which was the first step of our proof when ${A=B}$. But this argument collapses when ${B\not=A}$, since it happens that ${A_y}$ is empty and ${B_y}$ 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 ${\delta}$ and also volumes of superlevel sets of ${y\mapsto |A_y|}$.