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.

Motivation: Metabolomics has shown great potential to improve the understanding of complex diseases, potentially leading to therapeutic target identification. However, no single analytical method allows monitoring all metabolites in a sample, resulting in incomplete metabolic fingerprints. This incompleteness constitutes a stumbling block to interpretation, raising the need for methods that can enrich those fingerprints. We propose MetaboRank, a new solution inspired by social network recommendation systems for the identification of metabolites potentially related to a metabolic fingerprint. Results: MetaboRank method had been used to enrich metabolomics data obtained on cerebrospinal fluid samples from patients suffering from hepatic encephalopathy (HE). MetaboRank successfully recommended metabolites not present in the original fingerprint. The quality of recommendations was evaluated by using literature automatic search, in order to check that recommended metabolites could be related to the disease. Complementary mass spectrometry experiments and raw data analysis were performed to confirm these suggestions. In particular, MetaboRank recommended the overlooked α-ketoglutaramate as a metabolite which should be added to the metabolic fingerprint of HE, thus suggesting that metabolic fingerprints enhancement can provide new insight on complex diseases. Availability and implementation: Method is implemented in the MetExplore server and is available at www.metexplore.fr. A tutorial is available at https://metexplore.toulouse.inra.fr/com/tutorials/MetaboRank/2017-MetaboRank.pdf. Supplementary information: Supplementary data are available at Bioinformatics online.