Prékopa–Leindler inequality

Consider the following statements:

1. The shape with the largest volume enclosed by a given surface area is the $n$-dimensional sphere.
2. A marginal or sum of log-concave distributions is log-concave.
3. Any Lipschitz function of a standard $n$-dimensional Gaussian distribution concentrates around its mean.

What do these all have in common? Despite being fairly non-trivial and deep results, they all can be proved in less than half of a page using the Prékopa–Leindler inequality.

(I won’t show this here, or give formal versions of the statements above, but time permitting I will do so in a later blog post.)