The degree of approximation of sets in euclidean space using sets with bounded Vapnik-Chervonenkis dimension

The degree of approximation of infinite-dimensional function classes using finite n-dimensional manifolds has been the subject of a classical field of study in the area of mathematical approximation theory. In Ratsaby and Maiorov (1997), a new quantity ρ_n(F, L_q) which measures the degree of approximation of a function class F by the best manifold Hⁿ of pseudo-dimension less than or equal to n in the L_q-metric has been introduced. For sets F ⊂ ℝ^m it is defined as ρ_n(F, l^m_q) = inf_Hn. dist(F, Hⁿ), where dist(F, Hⁿ)- sup_x∈F inf_y∈Hn ∥x-y∥l^m_q and Hⁿ ⊂ ℝ^m is any set of VC-dimension less than or equal to n where n < m. It measures the degree of approximation of the set F by the optimal set Hⁿ ⊂ ℝ^m of VC-dimension less than or equal to n in the l^m_q-metric. In this paper we compute ρ_n(F, l^m_q) for F being the unit ball B^m_p = {x ∈ ℝ^m : ∥x∥l^m_p ≤ 1} for any 1 ≤ p, q ≤ ∞, and for F being any subset of the boolean m-cube of size larger than 2^mγ, for any 1/2 < γ < 1.