# 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.