Some Geometric Properties of Convex Bodies. Ii

Topological means are used for the study of approximation of 2-dimensional sections of a 3-dimensional convex body by affine-regular pentagons and approximation of a centrally symmetric convex body by a prism. Also, the problem of estimating the relative surface area of the sphere in a normed 3-space, the problem on universal covers for sets of unit diameter in Euclidean space, and some related questions are considered. Throughout, by a convex body K ⊂ R (a figure for n = 2) we mean a compact convex subset of R with nonempty interior. We denote by Gk(R) (respectively, G+k (R )) the Grassmann manifold of nonoriented (respectively, oriented) k-planes in R passing through O ∈ R. We let γ k : Ek(R ) → Gk(R) and (γ k ) : E k (R) → G + k (R ) be the tautological fiber bundles, where the fiber over an (oriented) k-plane α ∈ Gk(R) is α itself regarded as a k-dimensional vector space. We say that a field of convex bodies (or figures ; a CB or a CF-field) is given in a vector bundle γ if in each fiber α of γ we mark a convex body K(α) depending continuously on α. A CB-field is pointed if for each α we also mark a point x(α) ∈ K(α) depending continuously on α. (In other words, x(α) is a section of γ.) If λ ∈ R and P is an affine-regular polygon (e.g., a pentagon or a parallelogram) with center O(P ), then λP denotes the polygon homothetic to P with homothety ratio λ and homothety center O(P ). We denote by S(K) the area of a figure K ⊂ R. §1. Fields of convex figures in γ 2 and (γ 2), and 2-dimensional sections of convex bodies in R First, we prove two corollaries to the following known result. Theorem [2]. Each CF-field in γ 2 contains a figure circumscribed about an affine-regular octagon. Corollary 1. Suppose C is the bounded component of a cubic surface in R. Then each inner point O of C lies in a plane intersecting C along an ellipse. Proof. Indeed, C is convex automatically, because otherwise C intersects some line at 4 points. Consequently, the section of C by some 2-plane through O is circumscribed about an affine-regular octagon. Then this section is a component of a cubic, intersects an ellipse at 8 points, and, consequently, is an ellipse by the Bézout theorem. 2000 Mathematics Subject Classification. Primary 52A10, 52A15.