Grant: Discovery Grants Program - Individual

The proposed research project belongs to interconnected fields of Convex & Discrete Geometry, Geometric Tomography, and Information Theory. The principal focus is convex bodies in finite-dimensional Euclidean space, i.e., compact convex sets with non-empty interior, methods of their reproduction from given data, and associated probabilistic (entropic) liftings related to data transmission. The first goal of this proposal is the study of reconstruction methods for convex and discrete sets in the settings of non-central sections and point-projections using the tools of Harmonic Analysis, Integral Transforms, and the theory of Analytic Functions. A natural question of recognition of objects from various types of data about their sections, projections or both belongs to the core of Geometric Tomography. This direction is inspired by a rich set of pragmatic tasks such as scanning problems, X-ray procedures, radar searches, imaging processes of different scales. The notion of illumination interdependent with point-projections is one of the central settings in the related direction of work I am pursuing. It is inspired by the Levi-Hadwiger conjecture that asks about the least number of light sources needed to illuminate the surface of a convex body and classes that optimize this quantity. The question has been handled completely only for two-dimensional case, and a few classes of bodies under additional assumptions in higher dimensions. At the same time, according to recent results of Glazyrin, illumination almost everywhere with respect to Hausdorff measure, proposed by Bezdek, and Brass, Moser, Pach, has an unexpected outcome even in the case of a solid cube. I intend to establish exact lower and upper bounds of correct orders in terms of the dimension for several classes of convex bodies including, but not limited to, polytopes with sufficiently large number of symmetries and bodies of revolution. Another part of the proposed research is related to entropic analogues of the long-standing Kneser-Poulson conjecture that suggests that the volume of intersection of a finite family of balls should increase if the pairwise distances of centers decrease. Simultaneously, it is hypothesized that the volume of the union under such a contraction naturally should decrease. The conjecture is known to be true only for lower-dimensional settings and has strong relations to the problems of space packing, covering and reverse isoperimetric-type inequalities. I plan to extend my recent results with several peers that show an equivalence of corresponding questions for entropy and provide the affirmative answer for several classes of contractions and random variables. Furthermore, I am interested in the behaviour of other intrinsic volumes of convex bodies under contractions (for example, results in the spirit of the ones established earlier by Paouris and Pivovarov).