[Merged by Bors] - feature(Analysis/LocallyConvex/AbsConvex): Add zero_mem_absConvexHull#23845
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[Merged by Bors] - feature(Analysis/LocallyConvex/AbsConvex): Add zero_mem_absConvexHull#23845
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PR summary 587f1f5bc2Import changes for modified filesNo significant changes to the import graph Import changes for all files
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RemyDegenne
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| lemma zero_mem_absConvexHull {s : Set E} [SeminormedRing 𝕜] [NormOneClass 𝕜] | ||
| [AddCommGroup E] [Module ℝ E] [Module 𝕜 E] [Nonempty s] : 0 ∈ absConvexHull 𝕜 s := by | ||
| obtain ⟨w, hw⟩ := (inferInstance : Nonempty s) | ||
| rw [← add_neg_cancel ((1/2 : ℝ) • w), ← smul_neg] | ||
| exact convex_absConvexHull (subset_absConvexHull hw) | ||
| ((Balanced.neg_mem_iff balanced_absConvexHull).mpr (subset_absConvexHull hw)) | ||
| (le_of_lt one_half_pos) (le_of_lt one_half_pos) (add_halves 1) |
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absConvexHull is balanced, so you could use that 0 belongs to any nonempty balanced set (which seems to be a missing lemma):
lemma Balanced.zero_mem {E 𝕜 : Type*} [SeminormedRing 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s : Set E}
(hs_nonempty : s.Nonempty) (hs : Balanced 𝕜 s) : 0 ∈ s := by
have : (0 : E) = (0 : 𝕜) • (hs_nonempty.some : E) := by simp
rw [this]
exact hs.smul_mem (by simp) hs_nonempty.some_mem
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There is balancedCore_zero_mem and balancedCore_nonempty_iff with slightly different hypothesis.
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…#23845) The origin is a member of the absolute convex hull of a non-empty set (under appropriate hypothesis). Co-authored-by: Christopher Hoskin <[email protected]>
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…#23845) The origin is a member of the absolute convex hull of a non-empty set (under appropriate hypothesis). Co-authored-by: Christopher Hoskin <[email protected]>
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The origin is a member of the absolute convex hull of a non-empty set (under appropriate hypothesis).