Theorems · Theorem · convex and discrete geometry

PointedCone.IsFaceOf.mem_of_sum_smul_mem

∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : PartialOrder R] [inst_2 : IsOrderedRing R]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] {C F : PointedCone R M} {ι : Type u_4} [inst_5 : Fintype ι]
  {f : ι → M} {c : ι → R},
  F.IsFaceOf C → (∀ (i : ι), f i ∈ C) → (∀ (i : ι), 0 ≤ c i) → ∑ i, c i • f i ∈ F → ∀ (i : ι), 0 < c i → f i ∈ F

If the positive combination of points of a cone is in a face, then all the points are in the face.

Defined in

Mathlib.Geometry.Convex.Cone.Face.Basic

Cited by

0 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderIsOrderedRingAddCommGroupModuleFintype