Systems of geometric inequalities for relative measures of convex bodies in Minkowski spaces and corresponding extremal bodies
Final Report Abstract
During the one-year period, granted by the German Research Foundation, several parts of this (initially three-year) project were developed, and most of the implemented parts were accepted or at least submitted for publication. Some of points of the program are still in progress. The notion of reduced body K (the body whose minimum width decreases under any reduction, i.e., replacing K by a proper convex subset) can also be considered in Minkowski spaces (let us recall that a Minkowski space is a real normed linear space of finite dimension). Reduced bodies are essential with respect to a number of problems from convexity, namely the problems that involve thickness and extremal problems (or geometric inequalitise) for thickness. Jointly with H. Martini [AM] I obtained new information on Minkowskian reduced bodies. We were able to find some partial solutions to the following problem. M. Lassak (1990) asked whether reduced polytopes exist in the Euclidean space of dimension greater than two. We show that no Minkowskian reduced polytopes with d + 2 vertices (as well as no Minkowskian reduced polytope with d 4- 2 facets) exist under the assumption that the unit Minkowskian ball is regular. Furthermore, we also show that no simple polytope other than a simplex can be reduced in such kind of Minkowski spaces. As a consequence, we immediately get that in the Eucliden space of dimension greater than two no simple polytope is reduced. Further results involving classes of special convex bodies in Minkowski spaces were obtained in [Aveb], [AveOG] and [AMM]. I considered several problems dealing with finite point sets X and distances between them in general finite dimensional normed spaces. One tries to find relationships (in terms of inequalities) within this points sets in terms of distances that are involved in X as well as in terms of the cardinality of X and the dimension of the underlying Minkowski space. Problems of such kind turn out to be useful in bioinformatics, computer science and database design (in fact, in any area where pairwise quantitative comparison within large finite sets of objects is common). Jointly with Nico Düvelmeyer (TU Chemnitz) I developed new geometric methods of analysis of such kind of point sets. The new methods hinge upon geometric considerations that match well to tools from convex geometry. We confirm the general belief that the finite dimensional l∞, space must have the largest metric capacity (at least asymptotically). More suprisingly, we find some other spaces for which the maximum of the metric capacity is attained. The methods developed in [AD] will definitely be useful in the study of further open problems of a similar kind: for instance analysis of n-distance and equilateral sets, or the Menger type questions for Minkowski spaces. Other types of geometric ideas (involving linear optimziation) for such kind of problems were employed in [Avea]. Jointly with G. Bianchi [AB] I developed results on tomographic data (especially covariograms) and relationships within those data (some of which can expressed in terms of geometric inequalities.) Among other results obtained in [AB] we gave the following partial result to the well-known Matheron conjecture (1986). Planar convex bodies K with a local symmetry (pair of centrally symmetric boundary arcs) are determined by the covariogram data of K within the class of all planar convex bodies, up to translations and reflection. We mention that such kind of retrieval results are interesting for scientists working in mathematical physics on problems about quasi-periodic structures (the theory of quasicrystals). In particular, the obtained results have already attracted the attention of some of the experts in that area such as M. Baake (Bielefeld) and D. Lenz (Chemnitz). As a result of the project, I succeeded to perform a deep study of various aspects of geometric inequalities in Euclidean and Minkowski spaces. Furthermore, it was possible to provide more evidence that, in different forms, geometric inequalities for convex bodies and the correpsonding tools of obtaining them can appear in variuos applied disciplines of mathematics (e.g., see [AD] and [AB] as examples of application in geometric combinatorics and mathematical physics).
Publications
- On the inequality for volume and Minkowskian thickness, Canad. Math. Bull., 49, (2006),pp.l85-195
