This paper investigates a particular type of slice sampler algorithm, the polar slice sampler. This algorithm is shown to have convergence properties which are essentially independent of the dimension of the problem, at least for log-concave densities. For such densities, the algorithm provably converges (from appropriate starting point) to within 0:01 of stationarity in total variation distance in a number of iterations given as a computable function of the spherical asymmetry of the density. In particular, for spherically symmetric log-concave densities, in arbitrary dimension, with appropriate starting point, we prove that the algorithm converges in at most 525 iterations. Simulations are done which confirm the polar slice sampler's excellent performance. 1. Introduction.