Investigation of bounded convex bodies in R^n by probabilistic methods

Reconstruction of a convex body D from its probabilistic characteristics is one of the actual problems of stochastic geometry. In this context, there are various probabilistic characteristics used in literature and formulated in terms of random variables related to, for example, lower-dimensional cross sections of D or pairs of points taken from D. The corresponding problems are fundamental in geometric tomography and stereology with applications in medical diagnostics (see [1-2]). In particular, the problem of recognition of bounded convex domains by their chord length distribution function became a subject of active research due to G. Matheron’s hypothesis [3-4]. The hypothesis is examined in [5-10]. For recent advances and explicit computations of the chord length distribution function see [11-20]. For density function of the distance between two independent points chosen randomly and uniformly from D see [21-22]. The goal of this project is to make progress in the field of recognition of convex bodies in R^n based on the research conducted during recent years as well as to develop alternative probabilistic and statistical methods to establish new results in this direction. The field is challenging and promising in both theoretical and experimental means.