In this work we introduce a novel weighted message-passing algorithm based on the cavity method for estimating volume-related properties of random polytopes, properties which are relevant in various research fields ranging from metabolic networks, to neural networks, to compressed sensing. We propose, as opposed to adopting the usual approach consisting in approximating the real-valued cavity marginal distributions by a few parameters, using an algorithm to faithfully represent the entire marginal distribution. We explain various alternatives for implementing the algorithm and benchmarking the theoretical findings by showing concrete applications to random polytopes. The results obtained with our approach are found to be in very good agreement with the estimates produced by the Hit-and-Run algorithm, known to produce uniform sampling.
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