The BrunnMinkowski Inequality
The BrunnMinkowski Inequality gives an estimate for the measure of the set in terms of the Lebesgue measures of and in (of course, we can only speak of such an inequality if all the sets are Lebesgue measurable). The inequality has the form
First here are some facts that show that this is one of the only types of estimates we could expect. Since there are examples of sets with and we cannot bound above in terms of . For examples of such sets we have in ( is the Cantor set) and in more dimensions we can consider sets of the form such that .
Secondly, if we search for inequalities of the form
then by choosing convex and we would obtain the inequality
Since we have
we see that the best is .
Now let’s pass to the proof of the inequality. We consider of finite measure, or otherwise the inequality is just . If are dimensional boxes then the inequality becomes
Since this inequality does not change if we change to we can choose the factor such that . Then it becomes
which is an immediate consequence of the AMGM inequality.
Consider now and as finite union of boxes as above with disjoint interiors. We would like to prove the inequality by induction after the number of boxes in and . Note that we can translate and independently, and the inequality is the same, by using the invariance to translations of the Lebesgue measure. So we translate such that some hyperplane separates two boxes in , and denote and the intersection of with and , respectively. Now translate such that
where are defined in the same way as . Then we have and the number of boxes in and is at most , and the same inequality holds for the number of boxes in and . This allows us to apply the induction hypothesis and get
Consider now sets of finite measures and . Then we can find which are finite union of boxes such that and . Since , the inequality follows letting .
The case where are compact sets is proved using the previous case (note that compact implies compact). Denote and the same definition for , which make open sets. Since , the inequality for implies the inequality for as .
When are arbitrary measurable sets with finite measures and is measurable, then approximate from inside with compact sets, and taking the limit, the inequality follows.

May 9, 2012 at 10:04 pmMinkowski content and the Isoperimetric Inequality « Problems – Beni Bogoşel