I have to prove that, given $W$ open bounded, a cone function $C(x)=a+b|x-x_o|$ such that $x_o\notin W$ and $u\in C^0(\overline{W})$ such that $u=C$ on $\partial W$ and $L(u,W)=|b|$ (this is the Lipschitz function of $u$ in $W$), then $u=C$ in $W$.
The attempt of my professor. By contradiction suppose that $u(x)>C(x)$ for some $x\in W$. At first assume that $b\geq0$. Since $W$ is bounded and $x_o\notin C$, there exist $x^*,x^{**}$ belonging to $\partial W$ and to the same ray of $C$ through $x$ (the ray of $C$ through $x$ is the half-line $t\mapsto x_o+t(x-x_o), \ t\geq0$). Then $$ u(x)-u(x^*)=u(x)-C(x^*)>C(x)-C(x^*)=b|x-x^*|, $$ in contradiction with $L(u, W)=|b|$. If $b<0$, in a simil way we reach a contradiction by using the point $x^{**}$ in place of $x^*$. Finally if $u(x)<C(x)$ for some $x\in W$, the same arguments apply to $-u$ and $-C$ and then we conclude that $u=C$ in $W$.
I do not understand this proof. In particular I do not understand why he distinguishes the cases $b\geq0$ and $b<0$ and when $b\geq0$ he considers the point $x^*$ and when $b\geq0$ he considers the point $x^{**}$ in place of $x^*$. Can someone help me to explicit the implicated steps that are not present in this proof?
Thank You
P.S.: the referring book (article) is http://users.jyu.fi/~peanju/preprints/tour.pdf (p. 8-9)