Volumes of codimension one manifolds and a randomized polynomial-time algorithm for computing the surface volume of a smooth convex body

In this thesis we develop estimates for volumes of smooth boundary manifolds by deriving explicit relations between a heat flow across the surface and its volume. Applying these estimates together with existing results on randomised volume computation and rates of rapidly mixing chains on convex sets, we are able to derive a randomised algorithm for computing the surface volume of a smooth convex body M , that runs in time that is polynomial in the dimension d, a smoothness parameter (the “condition number”) 1/τ and inverse error 1/2. We also include a brief survey of known results related to computing the volume and surface area of a convex body. Chapter 1 The heat flow across codimension one manifolds and their volume