Is Lipschitz norm the other name for Lipschitz constant?

Question

I am seeing the term Lipschitz norm used in some papers and denoted by $$\|\cdot\|_{Lip}$$

Is it the other name for Lipschitz constant?

Answer 1

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.

Answer 2

\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.

Answer 3

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!