zero_mem_of_symmetric — Mathlib

English

The zero convex body coerces to the singleton set {0}.

Русский

Ноль выпуклого тела приводится к единичному множеству {0}.

LaTeX

$$$(0 : ConvexBody V) : Set V = \{0\}$$$

Lean4

/-- A convex body that is symmetric contains `0`. -/
theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : ∀ x ∈ K, -x ∈ K) : 0 ∈ K :=
  by
  obtain ⟨x, hx⟩ := K.nonempty
  rw [show 0 = (1 / 2 : ℝ) • x + (1 / 2 : ℝ) • (-x) by simp]
  apply convex_iff_forall_pos.mp K.convex hx (h_symm x hx)
  all_goals linarith

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