Rate of Convergence of the Discrete Polya Algorithm from Convex Sets. a Particular Case

In this work we deal with best approximation in `p , 1 < p ≤ ∞, n ≥ 2. For 1 < p < ∞, let hp denote the best `p -approximation to f ∈ R from a closed, convex subset K of R, f 6∈ K, and let h∗ be a best uniform approximation to f from K. In case that h∗ − f = (ρ1, ρ2, · · · , ρn), |ρj | = ρ for j = 1, 2, · · · , n, we show that the behavior of ‖hp − h∗‖ as p→∞ depends on a property of separation of the set K from the `∞-ball {x ∈ R : ‖x−f‖ ≤ ρ} at h∗ − f .