Abstract
As D’Arcy Thompson pointed out in his pioneering book on Growth and Form, “THERE IS AN IMPORTANT RELATIONSHIP BETWEEN THE FORM OR SHAPE OF A BIOLOGICAL STRUCTURE AND HIS FUNCTION”. Shape analysis deals with the statistical analysis of a family of “objects” in presence of stochastic fluctuations; stochastic geometry deals with the analysis of geometric aspects of “objects” subject to stochastic fluctuations. The scope of this presentation is to introduce relevant mathematical concepts and methods of shape analysis and of stochastic geometry, thus providing a guided tour in a selected bibliography. A relevant aspect of stochastic geometry is the analysis of the spatial structure of objects which are random in location and shape. In this case we may simply say that the mathematical interest is in spatial occupation. In various cases, described by specific examples (Birth-and-growth model; Boolean Model; a tumor growth model based on an inhomogeneous Boolean model), spatial occupation occurs via a random tessellation of the available space region. Hence a quantitative description of a random closed set can be obtained in terms of mean densities of volumes, surfaces, edges, vertices, etc., at the various Hausdorff dimensions.
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Capasso, V. (2003). Shape and Size in Biology and Medicine. In: Benci, V., Cerrai, P., Freguglia, P., Israel, G., Pellegrini, C. (eds) Determinism, Holism, and Complexity. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4947-2_14
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DOI: https://doi.org/10.1007/978-1-4757-4947-2_14
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