setInterior — Mathlib

English

The interior of a simplex is the set of affine combinations of vertices with weights strictly between 0 and 1.

Русский

interior симплекс — множество аффинных сочетаний вершин со весами строго между 0 и 1.

LaTeX

$$$ s.\mathrm{interior} = s.\mathrm{setInterior}(\{0< w_i < 1\}_{i}) $$$

Lean4

/-- The interior of a simplex is the set of points that can be expressed as an affine combination
of the vertices with weights strictly between 0 and 1. This is equivalent to the intrinsic
interior of the convex hull of the vertices. -/
protected def setInterior (I : Set k) {n : ℕ} (s : Simplex k P n) : Set P :=
  {p | ∃ w : Fin (n + 1) → k, (∑ i, w i = 1) ∧ (∀ i, w i ∈ I) ∧ Finset.univ.affineCombination k s.points w = p}

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