On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space ℜ with at leastn + 2 points;f,h1,h2:Z → ℜ continuous functions,h1,h2 strictly positive andP(x,z),x≔(x <Subscript>0</Subscript>,...,x <Subscript> n </Subscript>)<Superscript>τ</Superscript> ε ℜ<Superscript> n+1</Superscript>,z ε ℜ, a polynomial of degree at mostn. Consider a feasible setM ≔ {x ε ℜ<Superscript> n+1</Superscript>∣∀z εZ, −h <Subscript>2</Subscript>(z) ≤P(x, z)−f(z)≤h <Subscript>1</Subscript>(z)}. Here it is proved the null vector 0 of ℜ<Superscript> n+1</Superscript> belongs to the compact convex hull of the gradients ± (1,z,...,z <Superscript> n </Superscript>), wherez εZ are the index points in which the constraint functions are active for a givenx* ε M, if and only ifM is a singleton. Copyright Physica-Verlag 1998