Geometric Tomography and Harmonic Analysis (14w5085)
Alexander Koldobsky (University of Missouri)
Dmitry Ryabogin (Kent State University)
Vladyslav Yaskin (University of Alberta)
Artem Zvavitch (Kent State University)
The main goal of the proposed workshop at BIRS would be to bring together the leading experts and young researchers in the area to coordinate the main applications of Harmonic Analysis to Convex Geometry and Geometric Tomography and to develop new Fourier analytic approaches to the main open problems in the area. A special effort will be made to attract a number of graduate students (see the attached list of participants).
The format of the meeting will be as follows. We are planning to have two to three plenary lectures every day followed by shorter talks. In addition we would like to allocate time for open problem sessions and informal discussions.
The proposed workshop will focus on several topics in Geometric Tomography but will also have very close relation to Harmonic, Geometric and Functional Analysis. Fourier analytic methods proved useful in the study of sections and projections of convex bodies, volumes and surface area measures, embeddings of normed spaces in $L_p$ and other topics. Below we outline a few main directions of the workshop that represent research interests of the majority of the proposed participants.
1. Uniqueness results.
It is a classical result in Geometric Tomography that origin-symmetric star bodies are uniquely determined by the areas of their central sections. However, it is unknown whether other intrinsic volumes of central sections (e.g. perimeters in the 3-dimensional case) can be used to determine a symmetric body uniquely. So, one of the goals of this workshop is to study problems of unique determination of convex/star bodies from lower dimensional data. Of great interest is the problem of Barker and Larman which asks whether convex bodies that contain a sphere of radius t in their interiors are uniquely determined by the volumes of sections by hyperplanes tangent to the sphere. The problem is open even in the planar case. We also plan to discuss recent solutions of problems of Bonnesen and Klee on determination of convex sets from the volume of maximal hyperplane sections. We hope that the discussion may lead to a further development towards the solution of the Bonnesen problem on determination of convex bodies from the volumes of their maximal sections and projections in the odd dimensions.
2. Inequalities for sections and projections.
The Busemann-Petty problem asks whether origin-symmetric convex bodies with smaller central hyperplane sections necessarily have smaller volume. Surprisingly, the answer to the problem is ``Yes" in dimensions 2, 3 and 4, and ``No" in higher dimensions. The problem was posed in 1956 and completely solved in mid-nineties. There are, however, modifications and generalizations of this problem that are still open. One of them is the lower dimensional Busemann-Petty problem, which asks the same question as the original Busemann-Petty problem but for sections of lower dimensions in place of hyperplanes. Another well-known open problem is the slicing problem, which asks whether any convex body of volume one has a large enough section (larger than a fixed constant independent of anything). We are also interested in non-central versions of the Busemann-Petty problem and the Shephard problem, which is the analogue of the Busemann-Petty problem for projections instead of sections.
Using the Fourier approach, the new concepts of stability and separation in volume comparison problems have recently been introduced. This properties were established in the settings of the Busemann-Petty and Shephard problems, and, moreover, it has been found that stability and separation naturally lead to various hyperplane inequalities in the spirit of the famous hyperplane problem. In particular, the hyperplane section inequality holds for intersection bodies with an arbitrary measure in place of volume. It is an excellent time now for experts on the hyperplane conjecture to study this phenomenon more closely and find connections between different kinds of hyperplane inequalities.
4. Classes of convex/star bodies associated with projections/sections.
Many of the problems discussed above are directly related to the study of certain classes of convex or star bodies. For example, intersection bodies played a key role in the understanding of the Busemann-Petty problem. Projection bodies are at the heart of the Shephard problem. The relation between Zhang's and Koldobsky's generalizations of intersection bodies stands behind the lower dimensional Busemann-Petty problem. Among open problems is the relation between polar projection bodies and intersection bodies, which is a key to understanding the duality between sections and projections of convex bodies. Such problems also have their interpretations in the language of functional analysis. In particular, many of the classes mentioned above can be described in terms of embeddings of normed spaces into $L_p$.
5. Complex convex bodies
The theory of real convex bodies goes back to ancient times and continues to be a very active field now. However, the situation with complex convex bodies is different, as no systematic studies of these bodies had been carried out, and results appeared only occasionally. After a recent solution of the complex Busemann-Petty problem, new methods of studying complex convex bodies were found that made this direction quite active. This activity includes a comprehensive study of complex intersection bodies and related results for sections. However, the study of projections of complex convex bodies seems to be more difficult and requires a coordinated effort of experts in the field.
All these topics are being actively pursued at the present time and any progress in these directions would be a tremendous success of the workshop. We believe it is the right time for such a meeting to take place.