8.1 Continuation of Volume Estimation

In the past two lectures, we have been analyzing the construction of an f.p.r.a.s. to estimate the volume of a convex body K ⊆ R in the membership oracle model. We assumed a general upper bound on the mixing time of heat-bath ball walks, and used this assumption to analyze our construction. In this section, we prove the assumed bound. Recall that the Poincaré constant is defined by α = infφ:K→R R K hφdμ Varμ φ , where the local variance is hφ = 12 ∫