mg.metric geometry – 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 defense on $n$ and $C$.)

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