For we have

where .

There is an easy proof when the RHS is . We will adjust the original proof.

By replacing with , it suffices to assume on . Suppose and achieves maximum at 0.

Firstly, . Let be the function whose graph is the cone with vertex and base . Since , there exists a ball such that . Let be the function whose graph is cone with vertex and base . Then

Define is the length of the ray in direction tha lies in ,

is defined as the region bounded by , . This is because such that means that

Suppose a bounded region in with boundary , , then the volume of . With this fact in hand,

and

By the holder inequality

.

So

Gilbarg Trudinger’s book chapter 9. 9.1

Advertisements