Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

Question

The Assignment:

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

A hint my tutor gave me is to use the MVT, which is why I'm trying to get an expression I can use it on. Since $ K $is compact and $f$ is continuously differentiable, the derivative will be bounded since we're in $\mathbb{R}$. I have several problems, firstly I still have no clue how/where to use the MVT and don't know where the fact that K is convex becomes important. (I do know the definition of convex, though.)

I'd appreciate any help.

score 5 · Accepted Answer · answered Jun 30 '14 at 00:06

5

Hint

Consider two points $x,y\in K.$ Since $K$ is convex you have a segment joining $x$ and $y$ which is contained in $K.$ Now apply the mean value theorem and use that derivatives are bounded on the compact set $K.$

answered Jun 30 '14 at 00:06

mfl

29,399

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

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