Stable isoperimetric inequality

Eric Carlen from Rutgers gave a colloquium last week in which he bought up some curious questions and facts regarding the ‘stability’ of standard geometric inequalities such as the isoperimetric and Brunn-Minkowski inequality. To prevent myself from forgetting it, I’m dropping a short note on this matter here. Interestingly I was unable to locate any reference to this nor did I take any notes, hence this post is completely based on my recollection of a lunch five days ago.

–Many thanks to Marco Barchies, serval very high-quality references are located now. It turns out that starting with Fusco-Maggi-Pratelli ’06  which contains a full proof of the sharp bound, there has been a collective progress on shorter/different proofs and variations of the theorem made. See comments below!

As we all know, for sets in $\mathbb{R}^n$, the isoperimetric inequality is sharp only when the set is a round ball. Now what if it’s ‘almost sharp’? Do we always have to have a set that’s ‘close’ to a round sphere? What’s the appropriate sense of ‘closeness’ to use?

One might first attempt to use the Hausdorff distance:

$D(S, B_1(\bar{0})) = \inf_{t \in \mathbb{R}^n}\{D_H(S+t, B_1(\bar{0}))$.

However, we can easily see that, in dimension $3$ or higher, a ball of radius slightly small than $1$ with a long and thin finger sticking out would have volume $1$, surface volume $\varepsilon$ larger than that of the unit ball, but huge Hausdorff distance:

In the plane, however it’s a classical theorem that any region $S$ of area $\pi$ and perimeter $m_1(\partial S) \leq 2\pi +\varepsilon$ as $D(S, B_1(\bar{0})) \leq f(\varepsilon)$ where $f(\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$ (well, that $f$ is because I forgot the exact bound, but should be linear in $\varepsilon$).

So what distance should we consider in higher dimensions? Turns out the nature thing is the $L^1$ norm:

$D(S, B_1(\bar{0})) = \inf_{t\in \mathbb{R}^n} \mbox{vol}((S+t)\Delta B_1(\bar{0}))$

where $\Delta$ is the symmetric difference.

First we can see that this clearly solves our problem with the thin finger:

To simplify notation, let’s normalize our set $S \subseteq \mathbb{R}^n$ to have volume 1. Let $B_n$ denote the ball with n-dimensional volume 1 in $\mathbb{R}^n$ (note: not the unit ball). $p_n = \mbox{vol}_{n-1}(\partial B_n)$ be the ($n-1$ dimensional) measure of the boundary of $B_n$.

Now we have a relation $D(S, B_n)^2 \leq C_n (\mbox{vol}_{n-1}(\partial S) - p_n)$

As said in the talk (and I can’t find any source to verify), there was a result in the 90’s that $D(S, B_n)^4 \leq C_n (\mbox{vol}_{n-1}(\partial S) - p_n)$ and the square one is fairly recent. The sharp constant $C_n$ is still unknown (not that I care much about the actual constant).

At the first glance I find the square quite curious (I thought it should depend on the dimension maybe like $n/(n-1)$ or something, since we are comparing some n-dimensional volume with (n-1) dimensional volume), let’s see why we should expect square here:

Take the simplest case, if we have a n-dimensional unit cube $C_n$, how does the left and right hand side change when we perturbe it to a rectangle with small eccentricity?

As we can see, $D(R_\varepsilon, C_n)$ is roughly $p_n \varepsilon$. The new boundary consists of two faces with measure $1+\varepsilon$, two faces of measure $1-\varepsilon$ and $2 \times (n-2)$ faces with volume $1-\epsilon^2$. Hence the linear term cancels out and we are left with a change in the order of $\varepsilon^2$! (well, actually to keep the volume 1, we need to have $1-\varepsilon/(1+\varepsilon)$ instead of $1-\varepsilon$, but it would still give $\varepsilon^2$)

It’s not hard to see that ellipses with small eccentricity behaves like rectangles.

Hence the square here is actually sharp. One can check that no matter how many of the $n$ side-length you perturbe, as long as the volume stay the same (up to $O(\varepsilon^2)$) the linear term of the change in boundary measure always cancels out.

There is an analoge of this stability theorem for the Brunn-Minkowski inequality, i.e. Given two sets of volume $V_1, V_2$, if the sum set has volume only a little bit larger than that of two round balls with those volumes, are the sets $L^1$ close to round balls? I believe it’s said this is only known under some restrictions on the sets (such as convex), which is strange to me since non-convex sets would only make the inequality worse (meaning the sum set has larger volume), don’t they?

I just can’t think of what could possibly go wrong for non-convex sets…(Really hope to find some reference on that!)

Anyways, speaking of sum sets, the following question recently caught my imagination (pointed out to me by Percy Wong, thank him~ and I shall quote him ‘this might sound like probability, but it’s really geometry!’):

Given a set $T \subseteq l^2$ (or $\mathbb{R}^n$), we define two quantities:

$G(T)=\mathbb{E}(\sup_{p \in T} \Sigma p_i g_i)$ and

$B(T) = \mathbb{E}(\sup_{p \in T} \Sigma p_i \epsilon_i)$

where $\mathbb{E}$ is the expected value, $\{g_i\}$ are independent random variables with a standard normal distribution (mean 0, variance 1) and $\{\epsilon_i\}$ are independent Bernoulli random variables.

Question: Given any $T$, can we always find $T'$ such that

$T \subseteq T' + B_{l^1}(\bar{0}, c B(T))$ and

$G(T') \leq c B(T)$

To find out more about the question, see Chapter 4 of this book. By the way, I should mention that there is a \$5000 prize for this :-P

7 thoughts on “Stable isoperimetric inequality”

1. tushar says:

Soon to uploaded… videos and notes for

A. Almut Burchard’s mini course:

http://www.msri.org/web/msri/scientific/workshops/show/-/event/Wm572

Title: Symmetrization
Speaker: Almut Burchard

Abstract: These lectures give an introduction to rearrangement or symmetrization methods in geometric and functional analysis. Rearrangements are geometric manipulations that improve the “shape” of a body while preserving its “size.”

We will mainly consider Steiner symmetrization (which introduces a hyperplane of symmetry) and polarization (which pushes mass across a hyperplane towards the origin); these can be iterated to produce full rotational symmetry.

Starting from Steiner’s 1830’s “simple proof” of the isoperimetric inequality, we will discuss the role of symmetrization in geometric optimization problems (e.g., minimizing the electrostatic capacity and Santalo’s inequality in convex geometry), functional inequalities (the Sobolev and Hardy-Littlewood- Sobolev inequalities), differential equations (eigenvalue problems and Talenti’s elliptic estimate), and probability (inequalities for Brownian motion and related processes). The goal is to acquaint students with a useful tool that can greatly simplify problems (if it applies).

and
B. Alessio Figalli’s talk:

http://www.msri.org/web/msri/scientific/workshops/show/-/event/Wm573

Title: Optimal Transport and Applications to Functional Inequalities
Speaker: Alessio Figalli

Abstract: It is by now known that many important functional and geometric inequalities (such as isoperimetric or Brunn-Minkowski inequalities) can be proved/improved using optimal transportation techniques. In this talk I’ll review some of these results.

…stay tuned!

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• Interesting…Almut taught my second year calculus class…
Were you in that MSRI quantitative geometry program?! Was it cool? (you should blog about it!)

Sorry for not replying your e-mail again >.< He just mentioned that 'application' to Urbanski's conjecture at the end of the talk and dropped down the statement, at the time I thought I knew how to do it once we are in Hilbert space…But now when I think about it, it's becomes unclear.

I can't find this in any of his papers either, so maybe if we still can't figure that out we can try to e-mail him and see if he respond?

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2. tushar says:

sounds good to me!

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• Thanks for letting me know!

However, somehow I can’t open any of the links…Umm…Were they from a priviate server?

Perhaps if you just give me the author and titles of them I can locate the papers…

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• Ah, it works now…(Guess it was just a temporary server down)

Thanks again! So it’a indeed a quite active topic!~ (Great to know! I feel completely ignorant)

I have added a note after the first paragraph.

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