Theorems · Theorem · convex and discrete geometry

Finset.centerMass_mem_convexHull_of_nonpos

∀ {R : Type u_1} {E : Type u_3} {ι : Type u_5} [inst : Field R] [inst_1 : AddCommGroup E] [inst_2 : Module R E]
  {s : Set E} {w : ι → R} {z : ι → E} [inst_3 : LinearOrder R] [IsStrictOrderedRing R] (t : Finset ι),
  (∀ i ∈ t, w i ≤ 0) → ∑ i ∈ t, w i < 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ (convexHull R) s

A version of Finset.centerMass_mem_convexHull for when the weights are nonpositive.

Defined in

Mathlib.Analysis.Convex.Combination

Cited by

2 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldAddCommGroupModuleLinearOrderIsStrictOrderedRing