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The quantitative analysis of three-dimensional (3-D) shapes in terms of morphology and functionality is one of the most challenging problems in medical image analysis. This paper proposes a general methodology that aims at solving part of this problem. It introduces a nonparametric hierarchical partitioning approach that operates on any arbitrary 3-D shape described as a triangle mesh. It first extends the concept of basin districts to the case of curved spaces through a partitioning process on a valuation representing the main curvatures over a polyhedral support. A hierarchical construction of basin districts is obtained from a watershed transform. The speed of the front propagation on the polyhedral surface is controlled by the local characteristics of the surface geometry. As a prerequisite, a set of co-processing tools has been developed that operates directly on a triangulated domain. This includes classical signal processing tasks (e.g., re-sampling, filtering) on a polyhedral support performing a trade-off between accuracy and efficiency. The ability to provide an intrinsic shape partition from any triangular mesh is useful in a wide range of applications from accurate geometric modeling, and hierarchical shape dissection to robust mesh compression. Examples are presented in the paper to illustrate the principles and methodology.