•-Approximation minimization of convex functions in fixed dimension

The authors consider the problem of minimizing a convex function v(λ), over a closed convex set ℝ⊆ℝ^m, where for some fixed c∈ℝⁿ, an n×m matrix A of reals, and a set X⊆ℝ’n, they define v(λ)=max∈(c-Aλ)’T x:x∈X∈. The authors show the following result: if for any given λ∈ℝ, and any fixed •, 0∈•∈1, v(λ) can be evaluated by an •-approximation (strongly) polynomial combinatorial algorithm, then for a fixed dimension m, v*=v(λ*)=min∈v(λ): λ∈ℝ∈ can also be determined by an •-approximation (strongly) polynomial combinatorial algorithm.