∀ {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] {f : E → β} {a : E},
  IsLocalMin f a → ConvexOn ℝ Set.univ f → ∀ (x : E), f a ≤ f x