File: Mathlib/Analysis/NormedSpace/AddTorsorBases.lean

full_name: isOpenMap_barycentric_coord
code: theorem isOpenMap_barycentric_coord [Nontrivial ι] (b : AffineBasis ι 𝕜 P) (i : ι) :
    IsOpenMap (b.coord i) 
goal: IsOpenMap ⇑(b.coord i)

full_name: continuous_barycentric_coord
code: @[continuity]
theorem continuous_barycentric_coord (i : ι) : Continuous (b.coord i) 
goal: Continuous ⇑(b.coord i)

full_name: AffineBasis.interior_convexHull
code: /-- Given a finite-dimensional normed real vector space, the interior of the convex hull of an
affine basis is the set of points whose barycentric coordinates are strictly positive with respect
to this basis.

TODO Restate this result for affine spaces (instead of vector spaces) once the definition of
convexity is generalised to this setting. -/
theorem AffineBasis.interior_convexHull {ι E : Type*} [Finite ι] [NormedAddCommGroup E]
    [NormedSpace ℝ E] (b : AffineBasis ι ℝ E) :
    interior (convexHull ℝ (range b)) = {x | ∀ i, 0 < b.coord i x} 
goal: interior ((convexHull ℝ) (Set.range ⇑b)) = {x | ∀ (i : ι), 0 < (b.coord i) x}

full_name: IsOpen.exists_between_affineIndependent_span_eq_top
code: /-- Given a set `s` of affine-independent points belonging to an open set `u`, we may extend `s` to
an affine basis, all of whose elements belong to `u`. -/
theorem IsOpen.exists_between_affineIndependent_span_eq_top {s u : Set P} (hu : IsOpen u)
    (hsu : s ⊆ u) (hne : s.Nonempty) (h : AffineIndependent ℝ ((↑) : s → P)) :
    ∃ t : Set P, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ ((↑) : t → P) ∧ affineSpan ℝ t = ⊤ 
goal: ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤

full_name: IsOpen.exists_subset_affineIndependent_span_eq_top
code: theorem IsOpen.exists_subset_affineIndependent_span_eq_top {u : Set P} (hu : IsOpen u)
    (hne : u.Nonempty) : ∃ s ⊆ u, AffineIndependent ℝ ((↑) : s → P) ∧ affineSpan ℝ s = ⊤ 
goal: ∃ s ⊆ u, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤

full_name: IsOpen.affineSpan_eq_top
code: /-- The affine span of a nonempty open set is `⊤`. -/
theorem IsOpen.affineSpan_eq_top {u : Set P} (hu : IsOpen u) (hne : u.Nonempty) :
    affineSpan ℝ u = ⊤ 
goal: affineSpan ℝ u = ⊤

full_name: affineSpan_eq_top_of_nonempty_interior
code: theorem affineSpan_eq_top_of_nonempty_interior {s : Set V}
    (hs : (interior <| convexHull ℝ s).Nonempty) : affineSpan ℝ s = ⊤ 
goal: affineSpan ℝ s = ⊤

full_name: AffineBasis.centroid_mem_interior_convexHull
code: theorem AffineBasis.centroid_mem_interior_convexHull {ι} [Fintype ι] (b : AffineBasis ι ℝ V) :
    Finset.univ.centroid ℝ b ∈ interior (convexHull ℝ (range b)) 
goal: Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (Set.range ⇑b))

full_name: interior_convexHull_nonempty_iff_affineSpan_eq_top
code: theorem interior_convexHull_nonempty_iff_affineSpan_eq_top [FiniteDimensional ℝ V] {s : Set V} :
    (interior (convexHull ℝ s)).Nonempty ↔ affineSpan ℝ s = ⊤ 
goal: (interior ((convexHull ℝ) s)).Nonempty ↔ affineSpan ℝ s = ⊤

full_name: Convex.interior_nonempty_iff_affineSpan_eq_top
code: theorem Convex.interior_nonempty_iff_affineSpan_eq_top [FiniteDimensional ℝ V] {s : Set V}
    (hs : Convex ℝ s) : (interior s).Nonempty ↔ affineSpan ℝ s = ⊤ 
goal: (interior s).Nonempty ↔ affineSpan ℝ s = ⊤