Euclidean structure in finite dimensional normed spaces (Q2760183)

This survey article presents results that belong to a branch of mathematics which stands between convex geometry and functional analysis. This theory grows out of the classical Dvoretzky theorem that was one of initial results of the present local theory of Banach spaces, which uses as one of its main tools classical (and new) geometrical inequalities (such as isoperimetric inequalities, Brunn-Minkowski inequality, etc.). This direction of dependence of theories may be inverted and results in Banach space theory give a lot of information about structure of convex bodies. Namely, it was noted that every infinite-dimensional convex body (especially, the unit cell of a given Banach space) has a lot of hidden symmetries that led to new results in convex geometry in high finite dimensions and to build the so-called asymptotic theory.NEWLINENEWLINENEWLINEThe paper under review contains some background of convex geometry (Sections 2 and 3), a discussion of the Dvoretzky theorem including the concentration phenomenon and a discussion of certain Euclidean structures associated with a given norm or, equivalently, with a given convex body (Sections 4 and 5), applications to classical convex geometry (Section 6) and an Appendix that contains a short survey of the hyperplane conjecture, geometry of the Banach-Mazur compact (that is usually called the Minkowski compact), type-cotype theory and nonlinear type theory.NEWLINENEWLINENEWLINEThe article is highly recommended to all who are interested in convex geometry and its connections with Banach space theory.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].