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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | import Mathlib
namespace OAI
noncomputable section
open Set
namespace WeakMTWTransport
lemma finite_simplex_representation {E : Type*} [AddCommGroup E] [Module ℝ E]
[FiniteDimensional ℝ E] {S : Set E} {x : E} (hx : x ∈ convexHull ℝ S) :
∃ m : ℕ, 0 < m ∧ m ≤ Module.finrank ℝ E + 1 ∧
∃ z : Fin m → E, ∃ w : Fin m → ℝ,
(∀ i, z i ∈ S) ∧ (∀ i, 0 ≤ w i) ∧
(∑ i, w i = 1) ∧ (∑ i, w i • z i = x) := by
classical
obtain ⟨ι,inst,z,w,hz,hi,hw,hsum,heq⟩ := eq_pos_convex_span_of_mem_convexHull hx
let : Fintype ι := inst
have : Nonempty ι := by
by_contra h
have : IsEmpty ι := not_nonempty_iff.mp h
simp at hsum
let e := Fintype.equivFin ι
refine ⟨Fintype.card ι,Fintype.card_pos,
hi.card_le_finrank_succ.trans (Nat.add_le_add_right (Submodule.finrank_le _) 1),
fun i => z (e.symm i),fun i => w (e.symm i),?_,?_,?_,?_⟩
· intro i
exact hz (mem_range_self _)
· intro i
exact (hw _).le
· simpa only [e.symm.sum_comp] using hsum
· exact (e.symm.sum_comp (fun i => w i • z i)).trans heq
end WeakMTWTransport
end
end OAI
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