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. We present examples computed with our implementation in 2, 3 and 4 dimensions.

翻译：本文计算了由凹多项式不等式定义的凸体的体积。得益于周期通过线性微分方程的表示，这些体积可以达到任意精度。我们的方法基于Lairez、Mezzarobba和Safey El Din的工作。我们提出了一种新方法来识别相关临界值。凸性允许我们将所需的创造性伸缩步骤数量减少指数级。我们提供了基于SageMath中ore_algebra包的计算实现，并展示了在二维、三维和四维空间中使用该实现计算的示例。