Theorems · Theorem · geometry

Affine.Simplex.sSameSide_affineSpan_faceOpposite_of_sign_eq

∀ {R : Type u_1} {V : Type u_2} {P : Type u_4} [inst : Field R] [inst_1 : LinearOrder R]
  [inst_2 : IsStrictOrderedRing R] [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : AddTorsor V P] {n : ℕ}
  [inst_6 : NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R},
  ∑ j, w₁ j = 1 →
    ∑ j, w₂ j = 1 →
      ∀ {i : Fin (n + 1)},
        SignType.sign (w₁ i) = SignType.sign (w₂ i) →
          w₁ i ≠ 0 →
            (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide
              ((Finset.affineCombination R Finset.univ s.points) w₁)
              ((Finset.affineCombination R Finset.univ s.points) w₂)

Defined in

Mathlib.Analysis.Convex.Side

Cited by

3 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupModuleAddTorsorNeZero