an interior point of a convex set

Question

How can we prove a point is an interior point of a convex set, considering we don't have all of the extreme points of the given convex set ?

or

How can we find an interior point of a convex set, considering we don't have all of the extreme points of the given convex set ? (convex combination of those extreme points would not work as it may belong to a 'edge' of that convex set)

Answer 1

Since you are saying you don't know all the extreme points of the convex set, presumably you have something like an oracle that will tell you whether a point belongs to the set or not. Given such an oracle, choose a random orthonormal basis $\{v_i\}$ and choose a really small $\epsilon$, and then for your point $x$ in question query the oracle to determine if $x \pm \epsilon v_i$ is contained in the set for all $v_i$ in the basis. If it is for all $\pm v_i$, then your point is in the interior. Otherwise if not all these points are contained in your set, you can keep reducing $\epsilon$ until you are confident your point is on the boundary, or at least very very close to it. Note in order for this to work you need your convex set to be full-dimensional.

Answer 2

Empty sets are convex.

Moreover it depends on which topology you consider (e.g. $[0,1]\times \{0\}$ has empty interior in the Euclidean topology of $\mathbb{R}^2$, but non empty interior in the induced topology).