Theorems · Theorem · convex and discrete geometry

IsVisible.of_convexHull_of_pos

∀ {𝕜 : Type u_1} {V : Type u_2} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [inst_3 : AddCommGroup V] [inst_4 : Module 𝕜 V] {s : Set V} {x : V} {ι : Type u_4} {t : Finset ι} {a : ι → V}
  {w : ι → 𝕜},
  (∀ i ∈ t, 0 ≤ w i) →
    ∑ i ∈ t, w i = 1 →
      (∀ i ∈ t, a i ∈ s) →
        x ∉ (convexHull 𝕜) s →
          IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i) →
            ∀ {i : ι}, i ∈ t → 0 < w i → IsVisible 𝕜 ((convexHull 𝕜) s) x (a i)

If a point x sees a convex combination of points of a set s through convexHull ℝ s ∌ x, then it sees all terms of that combination. Note that the converse does not hold.

Defined in

Mathlib.Analysis.Convex.Visible

Cited by

1 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupModule