Lexicographical Representation of Convex Sets

We introduce two new families of properties on convex sets of $\mathbb{R}^n$, in order to establish new theorems regarding open and closed separation of a convex set from any outside point by linear operators from $\mathbb{R}^n$ to $\mathbb{R}^m$, in the sense of the lexicographical order of $\mathbb{R}^m$, for each m = 1, 2,..., n. We thus obtain lexicographical extensions of well known separation theorems for convex sets as well as characterizations of the solution sets of lexicographical (weak and strict) inequality systems defined by matrices of a given rank.