Approximation of plane cross sections of a convex body

Three theorems on approximation of plane sections of convex bodies by affine-regular polygons, ellipses, or circles are proved by topological means. In particular, it is proved that if K is a convex body in ℝ³ (resp., ℝ⁴), then for every interior point O of K there is a plane cross section of K through O which is circumscribed about an affine-regular hexagon (resp., octagon) with center O. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 174–183. Translated by N. Yu. Netsvetaev.     

Estimating the diameter of the space of plane convex figures with respect to an affine-invariant metric

A convex figure K ⊂ ℝ² is a compact convex set with nonempty interior, and αK is a homothetic image of K with coefficient α ∈ ℝ. It is proved that for any two convex figures K₁, K₂ ⊂ ℝ² there is an affine transformation T of the plane such that K₁ ⊂ T(K₂) ⊂ 2.7K₁. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 58–66.     

Affine images of the rhombo-dodecahedron that are circumscribed about a three-dimensional convex body

The main result of the paper is dual to the author's earlier theorem on the affine images of the cube-octahedron (the convex hull of the midpoints of edges of a cube) inscribed in a three-dimensional convex body. The rhombododecahedron is the polyhedron dual to the cube-octahedron. Let us call a convex body in Κ⊂ℝ³ exceptional if it contains a parallelogram P and is contained in a cylinder with directrix P. It is proved that any nonexceptional convex body is inscribed in an affine image of the rhombo-dodecahedron. The author does not know if the assertion is true for all three-dimensional convex bodies. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 191–195. Translated by N. Yu. Netsvetaev.     

Pentagons inscribed in a closed convex curve

Two theorems are proved. Let the points A₁, A₂, A₃, A₄, and A₅ be the vertices of a convex pentagon inscribed in an ellipse, let Κ⊂ℝ² be a convex figure, and let A₀ be a fixed distinguished point of its boundary ϖK. If the sum of any two of the neighboring angles of the pentagon A₁A₂A₃A₄A₅ is greater than π or the boundary ϖK is C⁴-smooth and has positive curvature, then some affine image of the pentagon A₁A₂A₃A₄A₅ is inscribed in K and has A₀ as the image of the vertex A₁. (This is not true for arbitrary pentagons incribed in an ellipse and for arbitrary convex figures.) Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 184–190. Translated by N. Yu. Netsvetaev.     

Estimating from Above the Perimeter of an Asymmetric Unit Disk in the Minkowski Plane

It is proved that each convex planar figure K ⊂ ℝ² contains a point O such that the perimeter of K computed with respect to the Minkowski distance function of the pair (K, O) is at most 9. If K is a triangle, this estimate is sharp. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 262–266.     

Classification of topologically conjugate affine mappings

We consider affine mappings from ℝ ⁿ into ℝ ⁿ , n ≥ 1. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from R into R and also for affine mappings from ℝ ⁿ into ℝ ⁿ , n > 1, having at least one fixed point and the nonperiodic linear part. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 134–139, January, 2009.     

On the topology of the Steiner symmetrization mapping

We prove that the Steiner symmetrization mapping on the hyperspace of convex bodies in ℝ² is soft and homeomorphic to a fibration in the bundle of Q-manifolds over any compact subset in the hyperspace of symmetric nonpolyhedral subsets. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 9–12.     

The only convex body with extremal distance from the ball is the simplex

We can extend the Banach-Mazur distance to be a distance between non-symmetric sets by allowing affine transformations instead of linear transformations. It was proved in [J] that for every convex bodyK we haved(K, D)≤n. It is proved that ifK is a convex body in ℝ ⁿ such thatd(K, D)=n, thenK is a simplex. This article is an M.Sc. thesis written under the supervision of E. Gluskin and V.D. Milman at Tel Aviv University. Partially supported by a G.I.F. grant.     

Equipartition of a continuous mass distribution

Here are three samples of results. (1) Let m be a finite (absolutely) continuous mass distribution in ℝ², and let ℓ = {ℓ₁, ..., ℓ₅ ⊂ ℝ²} be a quintuple of rays with common origin such that any two adjacent angles between them make a sum of at most π. Then an affine image of ℓ subdivides m into five parts with any prescribed ratios. (2) For each finite continuous mass distribution m in ℝⁿ, there exist n mutually orthogonal hyperplanes any two of which quarter m. (3) Let m and m′ be two finite continuous mass distributions in ℝRⁿ with common center of symmetry O. Then there exist n hyperplanes through O any two of which quarter both m and m′. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 92–106.     

Representation of functions of several variables by differences of convex functions

Let Dⁿ be a convex compact set in ℝⁿ. If a function admits a representation of the form f=g−h, where g and are convex and h is bounded from above, then there exists a representation of the same form which is “minimal” in some sense. A recurrent procedure converging to this minimal representation is described. For piecewise-linear functions f (in the cases n=1,2), finite algorithms giving minimal representations are found. A number of examples clarifying some unexpected effects are given. Some problems are formulated. Bibliography: 5 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 36–65. Translated by S. Yu. Pilyugin.     

On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

Fundamental rectangles of admissible lattices

Let Λ be a unimodular lattice in ℝ², μ a homogeneous minimum of Λ; let P(a,b)⊂ℝ² be a rectangle with vertices at the points (a,0), ...(0,b), P(a, b)+X its image under the translation by a vector X ∈ ℝ². We prove that there exists a sequence of positive numbers v₁<v₂<...<v_k<... with , such that for u>μ the rectangle P(u, v_k)+X contains T=S(P)+R points of Λ, where |R|<5; here S(P) is the area of the rectangle. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 82–89. Translated by O. A. Ivanov.     

On Analogs of Steklov Averages for Functions Defined on Convex Compact Sets in R m

For an arbitrary nondegenerate convex compact set K ⊂ ℝ^m we construct analogs of Steklov functions. We establish estimates for the seminorms in the space of continuous functions on K in terms of various moduli of continuity. Some uniform approximation aggregates for functions of two variables are also studied. Bibliography: 11 titles. Dedicated to the memory of Professor of St.Petersburg University Garald Isidorovich Natanson __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 3–20.     

Lorentzian similarity manifolds

An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝ ^m+2), where G = ℝ ^m+2 ⋊ (O(m+1, 1)×ℝ⁺). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S ^m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

Lower bounds for weak epsilon-nets and stair-convexity

A set N ⊂ ℝ ^d is called a weak ɛ-net (with respect to convex sets) for a finite X ⊂ ℝ ^d if N intersects every convex set C with |X ∩ C| ≥ ɛ|X|. For every fixed d ≥ 2 and every r ≥ 1 we construct sets X ⊂ ℝ ^d for which every weak 1/r -net has at least Ω(r log ^d−1 r) points; this is the first superlinear lower bound for weak ɛ-nets in a fixed dimension.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

Polygons inscribed in a closed curve and a three-dimensional convex body

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 2n-link polyline inscribed in γ and lying in a hyperplane. Let K ⊂ ℝ³ 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.     

On differentiable area-preserving maps of the plane

F: ℝ² → ℝ² is an almost-area-preserving map if: (a) F is a topological embedding, not necessarily surjective; and (b) there exists a constant s > 0 such that for every measurable set B, μ(F(B)) = sμ(B) where μ is the Lebesgue measure. We study when a differentiable map whose Jacobian determinant is nonzero constant to be an almost-area-preserving map. In particular, if for all z, the eigenvalues of the Jacobian matrix DF _z are constant, F is an almost-area-preserving map with convex image.     

On the existence and approximation of zeroes

In this paper we consider the problem of finding zeroes of a continuous functionf from a convex, compact subsetU of ℝ ⁿ to ℝ ⁿ . In the first part of the paper it is proved thatf has a computable zero iff:C ⁿ →ℝ ⁿ satisfies the nonparallel condition for any two antipodal points on bdC ⁿ, i.e. if for anyx∈bdC ⁿ ,f(x)≠αf(−x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bdC ⁿ to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes off. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extendingf:U→ℝ ⁿ to a function g from a cube containingU to ℝ ⁿ , the procedure can be applied to any continuous functionf without having any information about the global and local Brouwer degrees a priori.