Exact Volumes of Semi-algebraic Convex Bodies

The volume of a convex body is a quantity of interest in geometric statistics. It is often computed using the Monte Carlo method, which lacks precision and accuracy. When the convex body is a semi-algebraic set, the volume can be computed to arbitrary precision by considering its representation as a period of a rational function. Using the theory of holonomic functions and D-modules, we compute the volume by solving a uni-variate linear differential equation.

In this talk I present joint work with Nicolas Weiss on a method and implementation to compute the volume of a convex body given by finitely many polynomial inequalities. Our approach rests on work of Lairez, Mezzarobba, and Safey El Din. By restricting our class of semi-algebraic sets to convex bodies, we provide a complete algorithm that, in comparison to general semi-algebraic sets, has exponentially fewer steps.