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](https://mathoverflow.net/questions/391380/a-robust-version-of-harpers-theorem) 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 $D = 100C$.



Stability of isoperimetric inequality
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.


(I assume that you know how to solve it for a union of dijoint balls; otherwise you would not ask the question.)


