We propose a new measure defined on the shape interior and projected onto surface meshes when necessary. The measure is robust under geometric transformations and nuisance factors including topological distortions, pose changes, and occlusions. The measure captures non-local and non-linear interactions within the shape domain; nevertheless, the computations remain local and even linear. This is achieved via pooling multiple solutions to a Poisson equation with varying right-hand-side terms. We use the measure to address shape analysis problems in two and three dimensions. Toward this end, firstly, we exploit the evolution of its level curves and extract a probabilistic representation of the shape decomposition hierarchy. Then, in the context of a groupwise shape analysis task, we demonstrate how such a probabilistic structure enables us to select the task-dependent optimum from the set of possible hierarchies. Finally, we devise an unsupervised mesh segmentation algorithm which utilizes the proposed measure after projecting it to the mesh surface. Benchmark evaluation shows that the algorithm performs the best among the unsupervised algorithms and even performs comparably to supervised and groupwise segmentation algorithms.