# Stability of isoperimetric inequality

Let $$S$$ be subset of $$\mathbb{R}^n$$ with perimeter 1.

Isoperimetric inequality states that then the volume of $$S$$ is not greater than $$V_n$$,

where $$V_n$$ is the volume of a ball in $$\mathbb{R}^n$$ with perimeter 1.

Assume that $$C \cdot \text{[Volume of }S] \ge V_n$$, where $$C$$ is some constant.

Is it true that 99% of $$S$$ can be covered by union of constant number of balls with constant radius? (The constants can be depend on $$n$$ and $$C$$.)

P.S. The question is motivated by the similar question about boolean cube.

UPD: I can prove this statement for $$n=2$$: if $$F$$ belongs to $$\mathbb{R}^2$$, the perimeter of $$F$$ is equal to $$1$$ and its area is at least $$\frac{1}{C}$$ then 99% of $$F$$ can be covered by $$1000C$$ balls with radius $$1$$.

Indeed, consider $$D:=100C$$ connectivity components of $$F$$ with the greatest perimeter. Of course, they can be coveres by $$100C$$ balls with radius $$1$$. Our goal is to show that other connectivity components of $$F$$ cannot have total area $$\frac{1}{100C}$$ or bigger. Denote the set of non-covered components by $$\mathcal{H}$$. We can assume that all elements of $$\mathcal{H}$$ are balls---this is the worst case scenario. More over we can assume that all---or all except one---balls have the same (the greatest) perimeter (because $$(r+a)^2 + (l-a)^2 > r^2 + l^2$$ if $$r > l$$ and $$a>0$$). So, if $$|\mathcal{H}|=m$$ then we have $$m$$ balls with perimeter at most $$\frac{1}{D+m}$$, the total area of these balls is less than $$\frac{1}{100C}$$ since $$D = 100C$$.

We can assume that the set is a union of disjoint balls on a large distance from each other. Indeed, cut the space into cubes of small fixed size $$a$$. Shifting $$S$$ we can assume that the total area of intersection of $$S$$ with cutting hyperplanes is less then some constant that depends on $$a$$. Now take intersection of $$S$$ with each cube and exchange it to isolated ball with the same surface area.
• Do you mean that $a$ is depend on $n$ and $C$ only? What is it? May 2, 2021 at 7:27
• You choose $a$ so that each cube can be covered by a ball of the given radius. May 2, 2021 at 16:21
• Ok, thank you, Could you please explain this: "the total area of intersection of $S$ with cutting hyperplanes is less then some constant that depends on $a$? May 2, 2021 at 16:52
• @AlexeyMilovanov Let $A$ be average area of intersection of $S$ with the grid hyperplanes. Then $n\cdot\mathrm{vol}\,S=A\cdot a$. Plus, by shifting $S$ you can make the area of intersection to be less than $A$. May 2, 2021 at 17:38