On feasible sets defined through Chebyshev approximation

Let Z be a compact set of the real space R with at least n + 2 points; f, h1, h2: Z -&gt R continuous functions, h1, h2 strictly positive and P(x,z), x:= (x0, ..., xn)τ qq Rn+1, z qq R, a polynomial of degree at most n. Consider a feasible set M := {x qq Rn+1 | qqz qq Z, -h2(z) 1(z)}. Here it is proved the null vector 0 of Rn+1 belongs to the compact convex hull of the gradients &plusmn (1,z, ..., zn), where z qq Z are the index points in which the constraint functions are active for a given x* qq M, if and only if M is a singleton.