ABSTRACT. Suppose K is the intersection of a finite number of closed halfspaces in a Hilbert space X. Starting with any point x ∈ X, it is shown that the sequence of iterates {x_{n}} generated by Dykstra’s cyclic projections algorithm satisfies the inequality

||x_{n} − P_{K}(x)|| ≤ ρc^{n}

for all n, where P_{K}(x) is the nearest point in K to x, ρ is a constant, and 0 ≤ c < 1. In the case where K is the intersection of just two closed half-spaces, a stronger result is established: The sequence of iterates is either finite or satisfies ||x_{n} − P_{K}(x)|| ≤ c^{n−1}||x−P_{K}(x)|| for all n, where c is the cosine of the angle between the two functionals which define the half-spaces. Moreover, the constant c is the best possible. Applications are made to isotone and convex regression, and linear and quadratic programming.