Suppose we have a collection of sets and a set of linear functions where each being closed polyhedrons in , i.e., for some , and and . Moreover, for any , and , we have . Then we can define function on such that
We further assume is convex. We now state a theorem concerning the Lipschitz constant of .
Theorem 1 Under the above setting, given arbitrary norms , is -Lipschitz continuous with respect to and , i.e., for all ,
Proof: By letting , and . we see the inequality (1) is equivalent to
Using the property that is well-defined on for any , we have that there exists and corresponding such that
for all where . We also let and . These s have the property that
where is due to inequality (3), is due to the definition of operator norm and is using equality (4). This proves inequality (1).
2 thoughts on “Lipschitz constant of piecewise linear (vector valued) functions”
Is there any textbook that introduce this theorem? I need a real-valued version of this theorem to cite.
I really don’t know. Perhaps too obvious for pure mathematicians.