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8e4d533 | 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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | import Mathlib
namespace OAI
open Set MeasureTheory
namespace BorsukNine
open Set
noncomputable def simplexPoint {n : ℕ} (z : EuclideanSpace ℝ (Fin n)) : Fin (n+1) → ℝ :=
Fin.cases (1 - ∑ i, z i) (fun i => z i)
def simplexInterior (n : ℕ) : Set (EuclideanSpace ℝ (Fin n)) :=
{z | (∀ i, 0 < z i) ∧ ∑ i, z i < 1}
lemma simplexInterior_isOpen (n : ℕ) : IsOpen (simplexInterior n) := by
have h₁ : IsOpen {z : EuclideanSpace ℝ (Fin n) | ∀ i, 0 < z i} := by
rw [show {z : EuclideanSpace ℝ (Fin n) | ∀ i, 0 < z i} =
⋂ i, {z : EuclideanSpace ℝ (Fin n) | 0 < z i} from by ext; simp]
exact isOpen_iInter_of_finite (fun i => isOpen_lt continuous_const (EuclideanSpace.proj i).continuous)
exact h₁.inter (isOpen_lt (continuous_finsetSum Finset.univ
(fun i _ => (EuclideanSpace.proj i).continuous)) continuous_const)
lemma simplexInterior_nonempty (n : ℕ) : (simplexInterior n).Nonempty := by
refine ⟨WithLp.toLp 2 (fun _ : Fin n => (1:ℝ)/(n+1)),?_,?_⟩
· intro i
change 0 < (1:ℝ)/(n+1)
positivity
· change ∑ _i : Fin n, (1:ℝ)/(n+1) < 1
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
rw [mul_one_div]
exact (div_lt_one (by positivity)).mpr (by linarith)
lemma simplexPoint_pos {n : ℕ} {z : EuclideanSpace ℝ (Fin n)} (hz : z ∈ simplexInterior n) :
∀ i, 0 < simplexPoint z i := by
intro i
refine Fin.cases ?_ (fun j => ?_) i
· exact sub_pos.mpr hz.2
· exact hz.1 j
lemma simplexPoint_sum {n : ℕ} (z : EuclideanSpace ℝ (Fin n)) : ∑ i, simplexPoint z i = 1 := by
rw [Fin.sum_univ_succ]
simp only [simplexPoint,Fin.cases_zero,Fin.cases_succ]
ring
lemma simplexPoint_succ {n : ℕ} (z : EuclideanSpace ℝ (Fin n)) (i : Fin n) :
simplexPoint z i.succ = z i := rfl
end BorsukNine
namespace BorsukNine
open Set
open scoped RealInnerProductSpace
end BorsukNine
end OAI
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