Distance between two subsets of a unit-volume convex body

In multidimensional spaces we observe a variety of different phenomenas. Some of them might seem strange, for example, the volume of an $n$-dimensional ball of radius 2022 goes to 0 as $n$ tends to $\infty$. Even though two points in the $n$-dimensional unit cube could be at a distance of $\sqrt{n}$, two subsets of volume $\varepsilon$ could not be too far from each other – the distance between them is bounded above by $C\cdot\sqrt{|\ln(\varepsilon)}|$ for some constant $C$ independent of $\varepsilon$ and $n$. For standard simplexes and hyperoctahedrons(multidimensional octahedrons) we should replace $C\cdot\sqrt{|\ln(\varepsilon)}|$ with $C\cdot|\ln(\varepsilon)|$.
In our approach the key role is played by the isoperimetric problem: what is the minimal surface area that a body of a certain volume could have? (This problem could be considered in various different settings, for example, in the space $\mathbb{R}^n$, on the surface a sphere, in the space $\mathbb{R}^n$ with gaussian measure, or in a cube $(0; 1)^n$.)