A new nonparametric method is described to estimate the distribution of mass within spherical galaxies. The problem of estimating the mass, M(r), within radius r is converted into a problem of estimating a regression function nonparametrically, subject to shape restrictions. We represent the restrictions by the interception of quadratic cones and use the second-order cone programming to estimate the unknown parameters. We establish asymptotic results that are used to construct confidence intervals on M(r). We apply the technique to new kinematic data for four dwarf galaxies. Results indicate that dark matter dominates the stellar kinematics of these systems at all radii.