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 , 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:
However, we can easily see that, in dimension or higher, a ball of radius slightly small than with a long and thin finger sticking out would have volume , surface volume larger than that of the unit ball, but huge Hausdorff distance:
In the plane, however it’s a classical theorem that any region of area and perimeter as where as (well, that is because I forgot the exact bound, but should be linear in ).
So what distance should we consider in higher dimensions? Turns out the nature thing is the norm:
where 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 to have volume 1. Let denote the ball with n-dimensional volume 1 in (note: not the unit ball). be the ( dimensional) measure of the boundary of .
Now we have a relation
As said in the talk (and I can’t find any source to verify), there was a result in the 90’s that and the square one is fairly recent. The sharp constant 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 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 , how does the left and right hand side change when we perturbe it to a rectangle with small eccentricity?
As we can see, is roughly . The new boundary consists of two faces with measure , two faces of measure and faces with volume . Hence the linear term cancels out and we are left with a change in the order of ! (well, actually to keep the volume 1, we need to have instead of , but it would still give )
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 side-length you perturbe, as long as the volume stay the same (up to ) 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 , if the sum set has volume only a little bit larger than that of two round balls with those volumes, are the sets 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 (or ), we define two quantities:
where is the expected value, are independent random variables with a standard normal distribution (mean 0, variance 1) and are independent Bernoulli random variables.
Question: Given any , can we always find such that
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