Theorems · Theorem · geometry

Affine.Simplex.affineCombination_mem_interior_face_iff_pos

∀ {k : Type u_1} {V : Type u_2} {P : Type u_4} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [inst_3 : AddTorsor V P] [inst_4 : PartialOrder k] [IsOrderedAddMonoid k] {n : ℕ} (s : Affine.Simplex k P n)
  {fs : Finset (Fin (n + 1))} {m : ℕ} [NeZero m] (h : fs.card = m + 1) {w : Fin (n + 1) → k},
  ∑ i, w i = 1 →
    ((Finset.affineCombination k Finset.univ s.points) w ∈ (s.face h).interior ↔
      (∀ i ∈ fs, 0 < w i) ∧ ∀ i ∉ fs, w i = 0)

Defined in

Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic

Cited by

3 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsorPartialOrderIsOrderedAddMonoidNeZero