Is Lipschitz norm the other name for Lipschitz constant? I am seeing the term Lipschitz norm used in some papers and denoted by $$\|\cdot\|_{Lip}$$
Is it the other name for Lipschitz constant? 
 A: The Lipschitz constant is only a semi-norm, unless there is some boundary condition or some other additional condition. Constants have Lipschitz constant zero. 
In practice this means that the symbol you mention may mean either 
$$
\|u\|_{\mathrm{Lip}} := \sup_x\sup_{h\ne 0} \frac{|u(x+h)-u(x)|}{|h|},$$
or
$$
\|u\|_{\mathrm{Lip}} := \sup_x\sup_{h\ne 0} \frac{|u(x+h)-u(x)|}{|h|} + \|u\|_\infty.
$$
The first is sometimes called "homogeneous" and the second "inhomogeneous" norm, but this is Sobolev space terminology. 
A: \begin{align*}
[f]_{x,D}=\sup_{x,y\in D, x\ne y}\dfrac{|f(x)-f(y)|}{|x-y|^{\alpha}},~~~~0<\alpha\leq 1,
\end{align*}
is called unfiromly Holder continuous with exponent $\alpha$ in $D$, see Elliptic Partial Differential Equations of Second Order, David Gilberg/Neil S. Trudinger.
So it is $\alpha=1$ for Lipschitz case.
A: I think is related, but not the same. If you have $f\in C(X,\mathbb{R})$, then you can define the Lipschitz norm as:
$$||f||_{Lip}:=\sup_{x,y\in X,x\neq y}\frac{|f(x)-f(y)|}{d(x,y)}$$
with $d(x,y)$ the metric of $X$. As defined above, $||f||_{Lip}$ could be $+\infty$. The functions for which this isn't true are the Lipschitz functions. And more specifically, the least number $M$ for which $||f||_{Lip}\leq M$ is the Lipschitz constant!
