Let A be a C*-algebra with stable rank one. We show that the Cuntz semigroup of A satisfies Riesz interpolation. If A is also separable, it follows that the Cuntz semigroup of A has finite infima. This has far-reaching applications: 1. We confirm a conjecture of Blackadar and Handelman for unital C*-algebras with stable rank one: The (not necessarily lower semicontinuous) normalized dimension functions on A form a Choquet simplex. 2. We confirm the global Glimm halving conjecture for unital C*-algebras with stable rank one: For each natural number k, the C*-algebra A has no nonzero representations of dimension less than k if and only if there exists a morphism from the cone over the algebra of k-by-k matrices to A with full range. 3. We solve the rank problem for separable, unital (not necessarily simple) C*-algebras with stable rank one that have no finite-dimensional quotients: For every lower semicontinuous, strictly positive, affine function f on the Choquet simplex of normalized 2-quasitraces on A, there exists a positive element in the stabilization of A that has rank f. This is joint work with Ramon Antoine, Francesc Perera and Leonel Robert.