Theorems · Theorem · geometry

Affine.Simplex.disjoint_closedInterior_shift

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : PartialOrder k] [IsOrderedAddMonoid k]
  [inst_3 : AddCommGroup V] [inst_4 : AddTorsor V P] [inst_5 : Module k V] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n)
  (i : Fin (n + 1)) {x : k},
  x < 0 ∨ 1 < x → Disjoint s.closedInterior ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) x)

The base of a simplex shifted with parameter outside $[0, 1]$ does not intersect the closed interior.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingPartialOrderIsOrderedAddMonoidAddCommGroupAddTorsorModuleNeZero