卷 212, 编号 5 (2016)

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.

Let S be a family of convex hexagons whose vertices are the vertices of two negatively homothetic equilateral triangles with common center. It is proved that if K is a smooth convex figure, then K is circumscribed either about a hexagon in S or about two pentagons with vertices at the vertices of two hexagons in S. In the latter case, the sixth vertex of one of the hexagons lies outside K, while the sixth vertex of another one lies inside K.

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({K₁}, {K₂}) = inf{log(b/a)} for the classes {K₁}, {K₂} ∈ T of convex bodies K₁ and K₂, where a and b are positive reals such that there is a similarity transformation A with aA(K₁) ⊂ K₂ ⊂ bA(K₁). Let D₂ be the planar unit disk. If x > 0, then F_x denotes the set of planar convex figures K in F with d({D₂}, {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 prove that each convex body in ℝⁿ has n pairwise orthogonal affine diameters d₁, . . . , d_n such that it is possible to shift each of them through a linear combination of direction vectors of the diameters with smaller numbers so that their translates will intersect at their common midpoint.

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, . . . , f_m 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(f_m) 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, . . . , x_m) of nonnegative reals with nonzero sum there exists an m-tuple of “velocities” for the faces f₁, . . . , f_m such that the m-tuple (F(reg(f₁)), . . . , F(reg(fm))) is proportional to (x₁, . . . , x_m). Bibliography: 1 title.

A dendron is defined as a continuum (a nonempty, connected, compact Hausdorff space) in which every two distinct points have a separation point. It is proved that if a group G acts on a dendron D by homeomorphisms, then either D contains a G-invariant subset consisting of one or two points or G contains a free noncommutative subgroup and, furthermore, the action is strongly proximal.

We find sufficient conditions under which a finite connected graph has a spanning tree with the following property. There is a numbering of edges and an injective mapping of the set of all edges of the tree to the set of all pairs of different chords (i.e., edges of the graph not contained in the tree) such that for any pair of chords in the image of the mapping, the cycles containing one chord from the pair and containing no other chords intersect along an edge in the preimage, and, maybe, along other edges of the tree with smaller numbers. The problem of study of graphs that possess this property appeared in the process of study the (isotopic) classification problem of embeddings of graphs in the 3-space. Bibliography: 3 titles.

We prove that homotopy invariants of finite degree distinguish homotopy classes of maps of a connected compact CW-complex to a nilpotent connected CW-complex with finitely generated homotopy groups. Bibliography: 12 titles.

Let η be the Atiyah–Patodi–Singer invariant considered on smooth, compact, oriented, threedimensional submanifolds of ℝⁿ, and let A be an additive subgroup of ℝ. The problem of computing the degree of invariants of the form η mod A is examined. Here, the functional definition of invariants of finite degree is used. (A similar approach is used in S. S. Podkorytov’s paper “Quadratic property of the rational semicharacteristic.”) The main results are as follows. If 1 ∉ A, then the degree is infinite. If \( \frac{1}{3}\in A \), then the degree is equal to 1. Bibliography: 10 titles.