Maximizing non-linear concave functions in fixed dimension

Consider a convex set P in R^d and a piece wise polynomial concave function F: P→R. Let A be an algorithm that given a point x ∈ IR^d computes F(x) if x ∈ P, or returns a concave polynomial p such that p(x) <0 but for any y ∈ P, p(y) ⩾ 0. The author assumes that d is fixed and that all comparisons in A depend on the sign of polynomial functions of the input point. He shows that under these conditions, one can find max_P F in time which is polynomial in the number of arithmetic operations of A. Using this method he gives the first strongly polynomial algorithms for many nonlinear parametric problems in fixed dimension, such as the parametric max flow problem, the parametric minimum s-t distance, the parametric spanning tree problem and other problems. In addition he shows that in one dimension, the same result holds even if one only knows how to approximate the value of F. Specifically, if one can obtain an α-approximation for F(x) then one can α-approximate the value of maxF. He thus obtains the first polynomial approximation algorithms for many NP-hard problems such as the parametric Euclidean traveling salesman problem