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
Thus we can bound the term by
where is due to inequality (3), is due to the definition of operator norm and is using equality (4). This proves inequality (1).