Systems of geometric inequalities for relative measures of convex bodies in Minkowski spaces and corresponding extremal bodies

• Applicant: Professor Dr. Gennadiy Averkov
• Subject Area: Mathematics
• Term: from 2004 to 2007
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5438652

In the theory of convex sets there are many important results on various measures associated with convex bodies, such as diameter, minimal with, surface area, in- and circumradius. It is natural to extend isotropic measures, which are invariant with respect to motions, to anisotropic measures, which are only invariant with respect to translations. Usually the anisotropic meaures under considerations can be analogously introduced as special measures in Minkowski spaces (i.e., in finite dimensional Banach spaces over the real field). In the spirit of the famous (and not completely known) Blaschke diagram, we plan to investigate systems of geometric inequalities for relative measures of convex bodies in finite dimensional normed linear spaces. We also intend to give geometric descriptions of the bodies which are extremal regarding these inequalities. Some further aspects are also part of the program, namely: fundamental properties of Minkowskian measures, related optimization problems and various metrical properties of special classes of convex bodies in normed linear spaces. In addition to this pure research part of the project a comprehensive monograph under the title "Bodies of constant with in Minkowski spaces" will be written jointly with Prof. H. Martini. In particular this monograph will cover all new results and methods referring to this class of convex bodies.

• DFG Programme: Research Grants