How to estimate subgradient?

Question

consider a general convex function $f$ which is Lipschitz continuous over $X$, i.e., $\exists M > 0$ such that $$\left|f(x)-f(y)\right| \leq M\|x-y\|.$$ Here $X\subseteq R^n $is a closed convex set .Try to prove $$\|g(x)\| \leq M,\forall x\in X$$, here $g(x)$ is a subgradient i.e., g(x) \in \partial{f(x)}. The Lipschitiz constanct M of $f(x)$in $x$ is $lip(x)=\max\{\partial{f(x)}\}$,which shows the relationship between Lipschitiz constanct and subgradient locally.And how to extend this local property to the whole domain? Thanks.

Answer 1

By convexity we have the following,

$$\langle g(x), y-x\rangle\leq f(y)-f(x) \leq |f(y)-f(x)|\leq M|y-x|$$

and thus $\langle g(x), \vec{y-x}\rangle \leq M$ where $\vec{y-x}$ is a unit vector. We know by Cauchy Schwarz that $\langle g(x), \vec{y-x}\rangle\leq |g(x)||\vec{y-x}|=|g(x)|$ with equality iff $g(x)$ and $\vec{y-x}$ are linearly dependent. Pick $y$ such that $g(x)$ and $\vec{y-x}$ are linearly dependent, then $|g(x)|=\langle g(x), \vec{y-x}\rangle\leq M$.