Theorem: Let be a convex set.

(i) if .

(ii) if and only if is an interior point of .

We need the following definition: . This is often known as a Gauge or Minkowski functional.

Proof:

(i) If , then , so . This is the easy part. The converse holds here, if , then .

(ii) We first prove the “only if” part. Assume . Suppose is not an interior point of , i.e. there exists such that for all , for some .

We then have . Combining with the subadditive property of the gauge, we have . Rearranging, we get . By considering the various possibilities of the sign of , and using the positive homogeneity of the gauge, we can obtain a contradiction. For example, if , . Since as , this implies , a contradiction.

Conversely, if is an interior point of , for all there exists such that for all .

We have for all , for all . Since it is a “for all” quantifier, we can choose in particular , .

Then we have , which leads to and .

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