Geometric theory of Banach spaces. Part II. Geometry of the unit sphere

Interest in a geometrical approach to the study of Banach spaces is due to the following circumstance. Banach spaces have rich linear topological properties, which are extremely convenient in applications. However, the definition of a $B$-space is inseparably linked with a norm, that is, with a fixed geometrical object – the unit ball $D(B)=\{x\in B:\|x\|\leqslant 1\}$, whereas the linear topological properties depend (by definition) only on the topology of the space, that is, on a class of bounded convex bodies. Thus, we are naturally led to the question: what can be said about the linear topological properties of a space in isometric terms, that is, whilst remaining within the framework of a given norm?
The possibility of a productive investigation in this direction is essentially an infinite-dimensional situation, since in the finite-dimensional case the linear topology of a space is uniquely determined by the dimension. In view of the simplicity of the topological properties of $n$-dimensional spaces, the aim and fundamental object of investigation are geometrical (for example, the geometry of convex bodies). In the infinite-dimensional case topological questions give rise to enough concern. In this paper I follow tradition and give the main attention to results that lie in the topological channel, although it seems to me that an intrinsic study of the geometric object (an infinite-dimensional convex body) is no less interesting.