isoperimetric

The isoperimetric problem: among all closed curves in the plane with equal perimeter, which encloses the largest volume? It's well-known that this is the circle.

A differentiable manifold $M$ satisfies a $d$-dimensional isoperimetric inequality if for any open set $D \subset M$,

Here $d$ is called the 'isoperimetric dimension' of $M$. In general, an isoperimetric inequality is a lower bound on a surface-area-to-volume ratio. In Euclidean space, this inequality corresponds to the fact that any body with the same volume as the unit ball must have larger surface area.In this case $C$ is known precisely since it follows from the formulae for the surface area and volume of a ball.

Probability measures: generalizing volume to an arbitrary measure $\mu$, surface area is defined in terms of the infinitesimal change in volume as the set $D$ is expanded to a new set $D^*_\epsilon$ that includes all points within an infinitesimal distance $\epsilon$ of its previous boundary. The generalized isoperimetric problem is then: of all sets $D$ having a given measure $\mu(D)$, which has the smallest surface area $\frac{\partial \mu(D^*_\epsilon)}{\partial \epsilon}$?

• For Gaussian measure, the answer is not balls but half-spaces.

Concentration:

References:

• Isoperimetric inequality - Wikipedia
• Gaussian isoperimetric inequality - Wikipedia