The literature contains quite a few theorems on subdividing the volume of a convex body by a system of cones and on the possibility to circumscribe the body about a polytope of one type or another. See R. N. Karasev, “ Topological methods in combinatorial geometry ,” Russian Math. Surveys, 63 , No. 6, 1031–1078 (2008) for a survey of similar results. In the following, we also prove theorems of this kind. As a limit case, we obtain well-known theorems on inscribed polytopes.

The paper discusses the following question: up to which precision, the parallel projection of a three-dimensional convex body can be approximated by the same projection of a plane section of the body.

The surface area of a polyhedron in a normed space is defined as the sum of the areas of its faces, each divided by the area of the central section of the unit ball, parallel to the face. This functional naturally extends to convex bodies. In this paper, it is proved, in particular, that the surface area of the unit sphere in any three-dimensional normed space does not exceed 8. Bibliography: 2 titles.

Let T be the set of convex bodies in ℝ k , and let T be the set of their similarity classes. If k = 2, then F is written instead of T. Ametric d on T is defined by setting d({K 1 }, {K 2 }) = inf{log(b/a)} for the classes {K 1 }, {K 2 } ∈ T of convex bodies K 1 and K 2 , where a and b are positive reals such that there is a similarity transformation A with aA(K 1 ) ⊂ K 2 ⊂ bA(K 1 ). Let D 2 be the planar unit disk. If x > 0, then F x denotes the set of planar convex figures K in F with d({D 2 }, {K}) ≥ x. In addition, the sets T and F are equipped with the usual Hausdorff metric. It is proved that if y > log(sec(π/n)) ≥ x for a certain integer n greater than 2, then no mapping F x → F y is SO(2)-equivariant. Let M k denote the space of k-dimensional convex polyhedra and let M k (n) ⊂ M k be the space of polyhedra with at most n hyperfaces (vertices). It is proved that there are no SO(k)-equivariant continuous mappings M k (n + k) → M k (n). Let T s be the closed subspace of T formed by centrally symmetric bodies. Let T x denote the closed subspace of T formed by the bodies K with d(T s , {K}) ≥ x > 0. It is proved that for each positive y there exists a positive x such that no mapping T x → T y is SO(k)-equivariant. Bibliography: 3 titles.

We construct new polyhedra such that some of their similar or affine images can be inscribed in (or circumscribed about) every centrally symmetric convex body.

Is it true that any set of n + 1 points in R n can be isometrically embedded into any n-dimensional real normed apace? For n ≥ 3, the answer to this question is unknown to the author of this paper. For n = 2, it is clear that the answer is positive. For n = 3, the problem is reduced to the case where four points lie in a plane. A certain reduction is assigned for arbitrary n .

The main results are as follows. Let T be a Euclidean tetrahedron such that the ratio of lengths in each pair of edges of T is at least \( \left(\sqrt{8/3}+1\right)/3<0.878 \). Then each three-dimensional real normed space contains an isometrically embedded set of vertices of T . Let E be a three-dimensional normed space, and let x be a preassigned real number greater than \( \sqrt{2/3} \). Then E contains an affine image Π of a regular quadrangular pyramid such that the lateral edges of Π have equal length, the base edges of Π have equal length, the base diagonals also have equal length, and the ratio between the length of the lateral edges and the length of the base edges is equal to x. Bibliography: 5 titles.

The paper contains a survey of results about the possibility of inscribing convex polygons of particular types into a plane convex figure. It is proved that if K is a smooth convex figure, then K is circumscribed either about four different reflection-symmetric, convex, equilateral pentagons or about a regular pentagon.

We prove a number of statements concerning lattice packings of mirror symmetric or centrally symmetric convex bodies. This enables one to establish the existence of sufficiently dense lattice packings of any three-dimensional convex body of such type. The main result states that each three-dimensional, mirror symmetric, convex body admits a lattice packing with density at least 8/27. Furthermore, two basis vectors of the lattice generating the packing can be chosen parallel to the plane of symmetry of the body. The best result for centrally symmetric bodies was obtained by Edwin Smith (2005): Each three-dimensional, centrally symmetric, convex body admits a lattice packing with density greater than 0.53835. In the present paper, it is only proved that each such body admits a lattice packing with density \( \left(\sqrt{3}+\sqrt[4]{3/4}+1/2\right)/6>0.527 \). Bibliography: 5 titles.

Let M ⊂ ℝn be a convex polyhedron, i.e., the intersection of a finite number of closed half-spaces that is bounded and has nonempty interior. Let each hyperplane of the hyperfaces f1, . . . , fm of M move inwards M in a self-parallel fashion at a constant nonnegative speed (it is assumed that at least one face has nonzero speed). This yields a “shrinking” polyhedron. Let reg(f1), . . . , reg(fm) be the parts of M (with disjoint interiors) swept by the faces f1, . . . , fm during the “shrinking” process. The main result is as follows. Let F be a functional on the class of convex compact subsets in ℝn. It is assumed that F is nonnegative and continuous (with respect to the Hausdorff metric) and, furthermore, F(K) = 0 if and only if dim(K) < n. Then for each m-tuple (x1, . . . , xm) of nonnegative reals with nonzero sum there exists an m-tuple of “velocities” for the faces f1, . . . , fm such that the m-tuple (F(reg(f1)), . . . , F(reg(fm))) is proportional to (x1, . . . , xm). Bibliography: 1 title.

Is it true that every interior point of a three-dimensional convex body lies on its planar section with an inscribed regular hexagon and the center of a centrally symmetric convex body lies on a planar section with an inscribed regular octagon? In this paper, we prove these propositions for cylinders of a special type.

The paper proposes several theorems on dividing the area of a convex figure by a system of rays with a common initial point. These results include the previous results of this type. As a limiting case, we also obtained several theorems on inscribing a polygon of some type into a convex figure.

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in \( {\mathbb{R}^n} \),then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in \( {\mathbb{R}^n} \),then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.

Let X be an affine cross-polytope, i.e., the convex hull of n segments A 1 B 1,…, A n B n in \( {\mathbb{R}^n} \) that have a common midpoint O and do not lie in a hyperplane. The affine flag F(X) of X is the chain O ∈ L 1 ⊂⋯ ⊂ L n = \( {\mathbb{R}^n} \), where L k is the k-dimensional affine hull of the segments A 1 B 1,…, A k B k , k ≤ n. It is proved that each convex body K ⊂ \( {\mathbb{R}^n} \) is circumscribed about an affine cross-polytope X such that the flag F(X) satisfies the following condition for each k ∈{2,…, n}:the (k−1)-planes of support at A k and B k to the body L k ∩ K in the k-plane L k are parallel to L k −1.Each such X has volume at least V(K)/2n(n−1)/2. Bibliography: 5 titles.

The main results are as follows. Let K be a three-dimensional body of constant unit width, and let L be a line. Let T L (K) denote the set of all points where tangents of K parallel to L touch K. It is proved that for each L the curve T L (K) is rectifiable and has length at most \( \sqrt {2} \pi \); this estimate is sharp. Furthermore, there always exists a line L such that the length of the orthogonal projection of T L (K) to L is at most sin(π/10) + sin(π/20) < 0.4655. Bibliography: 2 titles.

The following conjecture is discussed: if K is a plane convex figure and T is a triangle of maximal area contained in K, then K is contained in \( \sqrt {5} \) T. It is shown that it suffices to check the conjecture in the case where K is a convex hexagon, but the conjecture is proved only in the case where K is a pentagon. Bibliography: 2 titles.

Let K be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that K is contained in a centrally symmetric hexagonal prism (or a parallelepiped) of volume \( {{4} \left/ {{\sqrt[3]{3} < 2.7734451}} \right.} \). This fact implies that space contains a lattice packing of translates of K with density \( {{{\sqrt[3]{3}}} \left/ {{4 > 0.36056}} \right.} \). Furthermore, K is contained in a parallelepiped of volume \( \frac{4}{3}{\left( {2 + \sqrt {3} } \right)^{{{2} \left/ {3} \right.}}} < 3.2080203 \). Bibliography: 6 titles.

Let \( \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) \) be the tautological vector bundle over the Grassmann manifold of 2-planes in \( {\mathbb{R}^3} \), where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of \( {\mathbb{R}^3} \) . A field of convex figures is given in γ 2 3 if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ 2 3 contains a figure K containing a centrally symmetric convex figure of area \( \left( {4 + 16\sqrt {2} } \right) \) S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area \( \left( {12\sqrt {2} - 8} \right) \) S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume \( \left( {36\sqrt {2} - 24} \right) \) V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.

Known well is the problem of finding configurations of points on the Euclidean sphere S n that can be put into one level surface of any continuous function on S n by a rotation of S n . The paper is devoted to various ways of transferring this problem to the ease of a normed space. Here is one of the results. Let f and g be two even continuous functions on an n -dimensional normed space E , and let f (0) < f ( x ) for all nonzero x ∈ E . Then E contains n unit vectors e 1 ,…, e n such that for any 1 ≤ i ≤ j ≤ n we have f ( e i + e j ): f ( e i − e j ) and g ( e i + e j ): g ( e i − e j ). Bibliography: 16 titles.

Here are samples of results obtained in the paper. Let γ be a centrally symmetric closed curve in ℝ n that does not contain its center of symmetry, O . Then γ is circumscribed about a square (with center O ), as well as about a rhombus (also with center O ) whose vertices split γ into parts of equal length. If n is odd, then there is a centrally symmetric equilateral 2 n -link polyline inscribed in γ and lying in a hyperplane. Let K ⊂ ℝ 3 be a convex body, and let x ∈ (0; 1). Then K is circumscribed about an affine-regular pentagonal prism P such that the ratio of the lateral edge l of P to the longest chord of K parallel to l is equal to x . Bibliography: 7 titles.