Characterization of Convexity Using Functions

Let Ω ⊆ RN be a domain with continuously or C1 differentiable boundary. A continuously differentiable function ρ : RN → R is called a defining function for Ω if

1. Ω = {x ∈ RN : ρ(x) < 0}; 2. cΩ = {x ∈ RN : ρ(x) > 0}; 3. ∇ρ(x) 6= 0 ∀x ∈ ∂Ω.