# Documentation

Mathlib.Analysis.Convex.StoneSeparation

# 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 {𝕜 : Type u_1} {E : Type u_2} [] [Module 𝕜 E] {p : E} {q : E} {u : E} {v : E} {x : E} {y : E} {z : E} (hz : z segment 𝕜 x y) (hu : u segment 𝕜 x p) (hv : v segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (↑() {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 ∈ [x, y].

theorem exists_convex_convex_compl_subset {𝕜 : Type u_1} {E : Type u_2} [] [Module 𝕜 E] {s : Set E} {t : Set E} (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
C, Convex 𝕜 C Convex 𝕜 C s C t C

Stone's Separation Theorem