closedBall_eq_segment — Mathlib

English

If hf: StrongConvexOn s m f and m > 0, then f is StrictConvexOn on s.

Русский

Если hf: StrongConvexOn s m f и m > 0, то f строго выпукла на s.

LaTeX

$$$$ \text{StrongConvexOn}(s,m,f) \land (0 < m) \Rightarrow \text{StrictConvexOn}(\mathbb{R},s,f). $$$$

Lean4

theorem closedBall_eq_segment (hε : 0 ≤ ε) : closedBall r ε = segment ℝ (r - ε) (r + ε) := by
  rw [closedBall_eq_Icc, segment_eq_Icc ((sub_le_self _ hε).trans <| le_add_of_nonneg_right hε)]

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