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

在凸集理论中，有许多关于凸体的各种度量的重要结果，如直径、最小半径、表面积、内圆半径和外圆半径。将关于运动不变的各向同性度量推广到只关于平移不变的各向异性度量是很自然的。通常，在Minkowski空间中(即在实域上的有限维Banach空间中)，所考虑的各向异性度量可以类似地引入为特殊的度量。在著名的(但不是完全已知的)Blaschke图的精神下，我们计划研究有限维赋范线性空间中凸体的相对测度的几何不等式组。我们还打算给出关于这些不等的极端的物体的几何描述。一些进一步的方面也是程序的一部分，即：Minkowskian测度的基本性质，相关的最优化问题以及赋范线性空间中特殊凸体的各种度量性质。除了该项目的这一纯研究部分外，还将与H.马蒂尼教授共同撰写一本题为《Minkowski空间中的常量的身体》的综合专著。特别是，这本专著将涵盖涉及这类凸体的所有新结果和方法。

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.