I am learing about the separation theorem :
separation between a (closed) convex set $X$ in $\mathbb R^n$ and a vector outside $X$.
I know about it but what are some examples where separation is used to prove other theorems relevant for optimization.
And what is the case of separating N disjoint convex cones.(theorem statement) Any links where i can get it.
To give you an example where they appear in optimization: Duality Theory is a concept which is based on separation arguments. For example the Fenchel-Moreau theorem uses separation arguments and is used to derive other theorems concerning Conjugate duality. Also algorithms like the Alternating Direction Method of Multipliers or the Primal-Dual Hybrid Gradient Method are therefore based on separation arguments. You should be able to find reference online, as these concepts are quite common.