Theorems · Theorem · convex and discrete geometry

Affine.Simplex.mem_interior_iff_sbtw

∀ {R : Type u_1} {V : Type u_2} {P : Type u_4} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : AddCommGroup V]
  [inst_3 : Module R V] [inst_4 : AddTorsor V P] [IsOrderedRing R] [IsDomain R] [Module.IsTorsionFree R V]
  {s : Affine.Simplex R P 1} {p : P}, p ∈ s.interior ↔ Sbtw R (s.points 0) p (s.points 1)

A point lies in the interior of a 1-simplex if and only if it lies strictly between its vertices.

Defined in

Mathlib.Analysis.Convex.Between

Cited by

1 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingPartialOrderAddCommGroupModuleAddTorsorIsOrderedRingIsDomainModule.IsTorsionFree