We compute the volumes of convex bodies that are given by inequalities of concave polynomials. These volumes are found to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on work of Lairez, Mezzarobba, and Safey El Din. We present a novel method to identify the relevant critical values. Convexity allows us to reduce the required number of creative telescoping steps by an exponential factor. We provide an implementation based on the ore_algebra package in SageMath. This is applied to a problem in geometric statistics, where the convex body is an intersection of $\ell_p$-balls.

翻译：本文计算了由凹多项式不等式给出的凸体体积。得益于通过线性微分方程表示的周期，这些体积可以计算到任意精度。我们的方法建立在Lairez、Mezzarobba和Safey El Din的工作基础上，提出了一种识别相关临界值的新方法。凸性使我们能够将所需创造性伸缩步骤的数量减少指数级。我们提供了基于SageMath中ore_algebra软件包的实现，并将其应用于几何统计中的一个问题，其中凸体是$\ell_p$-球的交集。