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.
(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 .