Theorems · Theorem · convex and discrete geometry
starConvex_iff_openSegment_subset
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : Module 𝕜 E] {x : E} {s : Set E} [ZeroLEOneClass 𝕜],
x ∈ s → (StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → openSegment 𝕜 x y ⊆ s)- Defined in
- Mathlib.Analysis.Convex.Star
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- PartialOrderstatement and proof · cited by 6,410
- ZeroLEOneClassstatement and proof · cited by 304
- openSegmentstatement · cited by 102
- StarConvexstatement · cited by 62
- starConvex_iff_segment_subsetproof · cited by 3
- openSegment_subset_iff_segment_subsetproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- convex_iff_openSegment_subsetproof · cited by 4