## Bound of norm on gradient implies bound on function

Suppose , , prove that where is a ball with radius 1.

We are going to use the following lemma.

**Lemma 7.16 on GT’s book. p162**

Let be convex and . Then

,

where , .

For any concentric with , , apply the lemma 7.6

Since is uniformly bounded for , then the above inequality implies that

Let , we get .

For any bounded domain with locally lipschitz boundary, this proposition is true. Because the locally boundary enable the domain to have locally star-shaped property, we can apply this proposition locally.

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