toWeakSpace_closedConvexHull_eq — Mathlib

English

Same as 52458, reiterating the equality of closures under weak space mapping for convex sets.

Русский

То же самое, повторяя равенство замыканий при отображении в слабое пространство для выпуклых множеств.

LaTeX

$$toWeakSpace_closure {s : Set E} (hs : Convex ℝ s) : (toWeakSpace 𝕜 E) '' (closure s) = closure (toWeakSpace 𝕜 E '' s)$$

Lean4

theorem toWeakSpace_closedConvexHull_eq {s : Set E} :
    (toWeakSpace 𝕜 E) '' (closedConvexHull 𝕜 s) = closedConvexHull 𝕜 (toWeakSpace 𝕜 E '' s) :=
  by
  rw [closedConvexHull_eq_closure_convexHull (𝕜 := 𝕜), ((convex_convexHull 𝕜 s).lift ℝ).toWeakSpace_closure _,
    closedConvexHull_eq_closure_convexHull]
  congr
  refine LinearMap.image_convexHull (toWeakSpace 𝕜 E).toLinearMap s

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