Theorems · Theorem · convex and discrete geometry
Convex.segment_subset
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : SMul 𝕜 E] {s : Set E}, Convex 𝕜 s → ∀ {x y : E}, x ∈ s → y ∈ s → segment 𝕜 x y ⊆ s- Defined in
- Mathlib.Analysis.Convex.Basic
- Cited by
- 22 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- PartialOrderstatement and proof · cited by 6,410
- Convexstatement and proof · cited by 551
- segmentstatement · cited by 120
- convex_iff_segment_subsetproof · cited by 6
Cited by22
Results whose statement or proof uses this declaration.
- segment_subset_convexHullproof · cited by 4
- Convex.openSegment_subsetproof · cited by 3
- Convex.mapsTo_lineMapproof · cited by 3
- ConvexOn.strictMonoOnproof · cited by 3
- IsMinOn.of_isLocalMinOn_of_convexOnproof · cited by 2
- HolderOnWith.of_le_of_leproof · cited by 2
- HasFDerivWithinAt.curveIntegral_segment_source'proof · cited by 2
- Convex.nontrivial_iff_nonempty_interiorproof · cited by 2
- Convex.isLittleO_pow_succproof · cited by 1
- eventually_riemannianEDist_le_edist_extChartAtproof · cited by 1
- Sion.exists_lt_iInf_of_lt_iInf_of_supproof · cited by 1
- convexJoin_subsetproof · cited by 1