Theorems · Theorem · convex and discrete geometry

IsMinOn.of_isLocalMinOn_of_convexOn

∀ {E : Type u_1} {β : Type u_2} [inst : AddCommGroup E] [inst_1 : TopologicalSpace E] [inst_2 : Module ℝ E]
  [IsTopologicalAddGroup E] [ContinuousSMul ℝ E] [inst_5 : AddCommGroup β] [inst_6 : PartialOrder β]
  [IsOrderedAddMonoid β] [inst_8 : Module ℝ β] [IsOrderedModule ℝ β] [PosSMulReflectLE ℝ β] {s : Set E} {f : E → β}
  {a : E}, a ∈ s → IsLocalMinOn f s a → ConvexOn ℝ s f → IsMinOn f s a

A local minimum of a convex function is a global minimum, restricted to a set s.

Defined in

Mathlib.Analysis.Convex.Extrema

Cited by

2 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupTopologicalSpaceModuleIsTopologicalAddGroupContinuousSMulAddCommGroupPartialOrderIsOrderedAddMonoidModuleIsOrderedModulePosSMulReflectLE