In $\mathbb{R}^n$, consider two compact subsets $A,\ B$. If $V$ is a volume form, then $$ V(A+B)^\frac{1}{n}\geq V(A)^\frac{1}{n} + V(B)^\frac{1}{n}\tag {1}$$ which is called Brunn-Minkowski inequality, where $sA+tB :=\{sx+ty|x\in A,\ y\in B\}$.
As an application, we have an isoperimetric inequality.
For $0<t<1$, by (1) together with arithmetric mean - geometric mean inequality, $$ V((1-t)A + tB)\geq V(A)^{1-t} V(B)^t \tag {2}$$
If $f,\ g,\ m$ are characteristic functions on $A,\ B,\ (1-t)A + tB$, then $$ m((1-t)x+ty)\geq f(x)^{1-t} g(y)^{t} \tag {3}$$ on $\mathbb{R}^n$. And $(2)$ can be rewritten by $$ \int m(a) da \geq \bigg(\int f(a)da\bigg)^{1-t}\bigg(\int g(a)da \bigg)^t \tag {4}$$
Here Prekopa-Leindler inequality is : Any functions satisfying (3) have (4).
Question : Is there an application of Prekopa-Leindler inequality ?
Thank you in advance.