Theorems · Theorem · convex and discrete geometry

ConvexOn.isBoundedUnder_abs

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {C : Set E} {f : E → ℝ},
  ConvexOn ℝ C f →
    ∀ {x₀ : E},
      C ∈ nhds x₀ →
        (Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (nhds x₀) |f| ↔
          Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (nhds x₀) f)

Defined in

Mathlib.Analysis.Convex.Continuous

Cited by

2 results in Mathlib

Foundations

Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace