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Stone's separation theorem #

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This file prove 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_convex_hull_triple {𝕜 : Type u_1} {E : Type u_2} [linear_ordered_field 𝕜] [add_comm_group E] [module 𝕜 E] {p q u v x y 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) ((convex_hull 𝕜) {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} [linear_ordered_field 𝕜] [add_comm_group E] [module 𝕜 E] {s t : set E} (hs : convex 𝕜 s) (ht : convex 𝕜 t) (hst : disjoint s t) :
(C : set E), convex 𝕜 C convex 𝕜 C s C t C

Stone's Separation Theorem