Integer lattice

Hello,

perhaps it is a little boring to bring up this problem once again but
I have some problems with it, yet.

Problem: Let f:Z^2-->[0,oo) be a function such that f(n,m)=(f(n+1,m)
+f(n-1,m)+f(n,m+1)+f(n,m-1))/4 then f is constant.

I have seen the following solution in a book with the
title...well...something with "newman" in the title, but however, here
is the solution stated there:
All such functions f form a locally compact convex cone. It suffices
to show that there is only one extreme point. Let T be the translation
(1,0) and S be the translation (0,1) one has
f=1/4*fT+1/4*fT^{-1}+1/4*fS+1/4*fS^{-1}
For an extreme point holds fT=xf and fs=yf and one gets x=y=1.

I don't understand this solution completely. Why is it sufficient to
show that there is only one extreme point? In my opinion every cone
has only one extreme point...?

Thanks,
S.
.

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