| 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) |
|
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| 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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|
