Dimension-independent spectral gap of polar slice sampling

Polar slice sampling, a Markov chain construction for approximate sampling, performs, under suitable assumptions on the target and initial distribution, provably independent of the state space dimension. We extend the aforementioned result of Roberts & Rosenthal (2002) by proving a lower bound of 1/2 on the spectral gap for arbitrary dimension if the target density is rotationally invariant, log-concave along rays emanating from the origin and sufficiently smooth. The main tool along this path is the development of conditions, in terms of a generalized level set function, that imply an explicit lower bound on the spectral gap in a general slice sampling context. The latter result is potentially applicable beyond the polar slice sampling framework.