Brunn–Minkowski inequality (${\lambda }_n(A+B{)}^{1/n}\ge {\lambda }_n(A{)}^{1/n}+{\lambda }_n(B{)}^{1/n}$)

For nonempty open sets $A,B\subset {\mathbb{R}}^n$, the Lebesgue measure of their Minkowski sum satisfiesIt also holds for compact sets and in standard measurable-set formulations with the appropriate measurability qualification. Normalize both volumes to one and apply the Prékopa–Leindler inequality to indicator functions.