nontrivial_iff_nonempty_interior — Mathlib

English

If s is convex and x ∈ interior s, then closure s ⊆ interior (image of interior s under homothety about x with scale t>1).

Русский

Если s выпукло и x ∈ interior s, то closure(s) ⊆ interior( Homothety_x,t'' interior(s) ).

LaTeX

$$closure s ⊆ interior (homothety x t '' interior s)$$

Lean4

theorem nontrivial_iff_nonempty_interior {s : Set 𝕜} (hs : Convex 𝕜 s) : s.Nontrivial ↔ (interior s).Nonempty :=
  by
  constructor
  · rintro ⟨x, hx, y, hy, h⟩
    have hs' := Nonempty.mono <| interior_mono <| hs.segment_subset hx hy
    rw [segment_eq_Icc', interior_Icc, nonempty_Ioo, inf_lt_sup] at hs'
    exact hs' h
  · rintro ⟨x, hx⟩
    rcases eq_singleton_or_nontrivial (interior_subset hx) with rfl | h
    · rw [interior_singleton] at hx
      exact hx.elim
    · exact h

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