We improve the inequality used in Pronzato [2003. Removing non-optimal support points in D-optimum design algorithms. Statist. Probab. Lett. 63, 223–228] to remove points from the design space during the search for a D-optimum design. Let ξ be any design on a compact space X⊂Rm with a nonsingular information matrix, and let m+ε be the maximum of the variance function d(ξ,x) over all x∈X. We prove that any support point x* of a D-optimum design on X must satisfy the inequality d(ξ,x*)⩾m(1+ε/2-ε(4+ε-4/m)/2). We show that this new lower bound on d(ξ,x*) is, in a sense, the best possible, and how it can be used to accelerate algorithms for D-optimum design.