# Stone's separation theorem #

This file proves Stone's separation theorem. This tells us that any two disjoint convex sets can be separated by a convex set whose complement is also convex.

In locally convex real topological vector spaces, the Hahn-Banach separation theorems provide stronger statements: one may find a separating hyperplane, instead of merely a convex set whose complement is convex.

theorem not_disjoint_segment_convexHull_triple {๐ : Type u_1} {E : Type u_2} [] [] [Module ๐ E] {p : E} {q : E} {u : E} {v : E} {x : E} {y : E} {z : E} (hz : z โ segment ๐ x y) (hu : u โ segment ๐ x p) (hv : v โ segment ๐ y q) :
ยฌDisjoint (segment ๐ u v) ((convexHull ๐) {p, q, z})

In a tetrahedron with vertices x, y, p, q, any segment [u, v] joining the opposite edges [x, p] and [y, q] passes through any triangle of vertices p, q, z where z โ [x, y].

theorem exists_convex_convex_compl_subset {๐ : Type u_1} {E : Type u_2} [] [] [Module ๐ E] {s : Set E} {t : Set E} (hs : Convex ๐ s) (ht : Convex ๐ t) (hst : Disjoint s t) :
โ (C : Set E), Convex ๐ C โง Convex ๐ Cแถ โง s โ C โง t โ Cแถ

Stone's Separation Theorem