I am new to concept of extremal Lipschitz functions and I have several basic question I'm still unsure about.
To fix notation let $(X,d)$ be a metric space, $Lip(X)$ Banach space of Lipschitz function on $X$ with the norm $\|f\|=\sup_{x\neq y}\frac{\|f(x)-f(y)\|}{d(x,y)}$, $PLip(X)=Lip(X)/\{const\}$, and $B_1$ unit ball in $PLip(X)$. We say that $f \in B_1$ is extremal if it is not a convex combination of other functions from $B_1$.
$1$. Is $f$ being extremal equivalent to saying that there is no non-constant function $g$ such that $f+g,f-g \in B_1$? Does $g$ have to be in $B_1$?
$2$. Are distance functions, i.e. functions of the form $\pm d(x_0,x)$, always extremal?
$3$. What is the characterization of spaces for which the only extremal functions are distance functions?
$4$. What is a good introductory textbook that covers this kind of Lipschitzian analysis?