English
The convex hull of s equals zero if and only if s equals zero.
Русский
Выпуклая оболочка множества равна нулю тогда и только тогда, когда множество равно нулю.
LaTeX
$$$ convexHull 𝕜 s = 0 \\iff s = 0 $$$
Lean4
theorem image_convexHull (f : E →ᵃ[𝕜] F) (s : Set E) : f '' convexHull 𝕜 s = convexHull 𝕜 (f '' s) :=
by
apply Set.Subset.antisymm
· rw [Set.image_subset_iff]
refine convexHull_min ?_ ((convex_convexHull 𝕜 (f '' s)).affine_preimage f)
rw [← Set.image_subset_iff]
exact subset_convexHull 𝕜 (f '' s)
· exact convexHull_min (Set.image_mono (subset_convexHull 𝕜 s)) ((convex_convexHull 𝕜 s).affine_image f)