Heat Flow and a Faster Algorithm to Compute the Surface Area of a Convex Body

We draw on the observation that the amount of heat diffusing outside of a heated body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a convex body given by a membership oracle. Our method has a complexity of O*(n⁴), where n is the dimension, compared to O*( n^8.5) for the previous best algorithm. We show that our complexity cannot be improved given the current state-of-the-art in volume estimation