Let , we define the sum
The following properties are easy to prove and will be used in the proof of the Brunn-Minkowski inequality:
- If is open, then for any , is also open.
- If are bounded, then is bounded.
- If , , where are intervals, then .
Proof: Suppose first and are of the form
where are open bounded intervals. Then
where and are the lengths of and respectively. Therefore by the AM-GM inequality,
Now suppose and are disjoint unions of open (-dimensional) rectangular blocks. If , the inequality has been proved. So, for induction purpose, assume . We can wlog assume . Take a plane parallel to one of the coordinates planes, say , which separates and . Then separates into the disjoint union of and :
with . Suppose . Take a plane parallel to the plane which separates into and , with . So now
with . Note that and , and so we can apply the induction hypothesis to conclude that
On the other hand, it is easy to see that and are disjoint (they are separated by ), and hence
In the general case, take a sequence of (monotone increasing) open sets with , (i.e. and ), such that , are of the form considered in the previous case. By applying the previous argument to , and taking , we can get the result.
Remark 1 (Some remarks about the -dimensional balls and spheres.) Let be the gamma function defined by
It is known that has the following properties:
and the -dimensional volume of the -dimensional unit ball is
For example, , and .
Theorem 3 (Isoperimetric inequality) For any smooth closed hypersurface in , we have
where is the region enclosed by .
Proof: Let be the “parallel” hypersurface, where is the inward unit normal of at . For small , is a smooth hypersurface which bounds a region . Let be the open ball with radius (centered at ). Then it can be seen that . Therefore by the Brunn-Minkowski inequality (Theorem 2), we have (we’ll write and , which shouldn’t cause confusion.)
This is equivalent to the isoperimetric inequality by Remark 1.
Remark 2 I believe a more refined version of the above argument can be used to prove some inequalities involving the so called curvature measures. I will try to work it out later.