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 0 ,...,x n ) τ ε ℜ n+1 ,z ε ℜ, a polynomial of degree at mostn. Consider a feasible setM ≔ {x ε ℜ n+1 ∣∀z εZ, −h 2 (z) ≤P(x, z)−f(z)≤h 1 (z)}. Here it is proved the null vector 0 of ℜ n+1 belongs to the compact convex hull of the gradients ± (1,z,...,z n ), 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