Affine-inscribed and affine-circumscribed polygons and polytopes

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ³, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ³ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298. Translated by B. M. Bekker.     

On intersections of affine diameters of a convex body

The chord of a body is called an affine diameter if its ends belong to parallel, supporting hyperplanes of the body. Theorem: For any affine diameter AB of a convex body K Rⁿ there exists a pair of affine diameters of the body K, each of which intersects AB, forming between them a preassigned angle. For any two non-intersecting affine diameters of a three-dimensional convex body, there exists a third affine diameter intersecting them.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 70–73, 1990.     

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.     

The affine image of a convex body of constant breadth

A convex body is said to have constant diagonal if and only if the main diagonal of the circumscribed boxes has constant length. It is shown that ann-dimensional convex body,n≧3, is the affine image of a body of constant breadth if and only if it has constant diagonal. Affine images of bodies of constant breadth are also characterized by the property that the orthogonal projection on each hyperplane is the affine image of a body of constant breadth in that hyperplane.     

A stability result for a volume ratio

For a convex body K in ℝⁿ, the volume quotient is the ratio of the smallest volume of the circumscribed ellipsoids to the largest volume of the inscribed ellipsoids, raised to power 1/n. It attains its maximum if and only if K is a simplex. We improve this result by estimating the Banach-Mazur distance of K from a simplex if the volume quotient of K is close to the maximum. This work was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.     

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.     

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.     

Affine Diameters and Chords of Convex Compacta

Existence theorems are proved for collections of affine diameters of a convex body that satisfy various additional conditions. The area of a convex planar figure is estimated via the maximal length of a chord dividing the area of the figure in a given ratio. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 252–261.     

The affine complete hypersurfaces of constant Gauss-Kronecker curvature

Given a bounded convex domain Ω with C∞ boundary and a function ψ∈C∞（δΩ）, Li-Simon-Chen can construct an Euclidean complete and W-complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer.     

Enclosed convex hypersurfaces with maximal affine area

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C^1,α regularity for general domains and C^1,1 regularity if the domain is uniformly convex. This work is supported by the Australian Research Council. Research of Sheng was also supported by ZNSFC No. 102033. On leave from Zhejiang University.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

The Bernstein problem for affine maximal hypersurfaces

In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, R ², must be a paraboloid. More generally, we shall consider the n-dimensional case, R ⁿ , showing that the corresponding result holds in higher dimensions provided that a uniform, “strict convexity” condition holds. We also extend the notion of “affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension n≥10. Oblatum 16-IV-1999 & 4-XI-1999?Published online: 21 February 2000     

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 Lipschitzian properties of polyhedral multifunctions

In this paper, we show that for a polyhedral multifunctionF:R ⁿ →R ^m with convex range, the inverse functionF ⁻¹ is locally lower Lipschitzian at every point of the range ofF (equivalently Lipschitzian on the range ofF) if and only if the functionF is open. As a consequence, we show that for a piecewise affine functionf:R ⁿ →R ⁿ ,f is surjective andf ⁻¹ is Lipschitzian if and only iff is coherently oriented. An application, via Robinson's normal map formulation, leads to the following result in the context of affine variational inequalities: the solution mapping (as a function of the data vector) is nonempty-valued and Lipschitzian on the entire space if and only if the solution mapping is single-valued. This extends a recent result of Murthy, Parthasarathy and Sabatini, proved in the setting of linear complementarity problems. Research supported by the National Science Foundation Grant CCR-9307685.     

Convex surfaces which intersect each congruent copy of themselves in a connected set

LetK ⊂R ³ be a three-dimensional convex body such that, for every isometry ρ ofR ³, the boundaries ofK and ρK meet in a connected set. ThenK is a parallel set of some possibly degenerate linesegment.     

Jacobi Structures on Affine Bundles

Abstract We study affine Jacobi structures （brackets） on an affine bundle π ： A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A^＋ = ∪p∈M Aff（Ap, R） of affine functionals. In the case rank A = 0, it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A^＋. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited （strongly-affine or affine-homogeneous Jacobi structures） over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra.     

Reduced bodies in normed planes

We say that a convex body R of a d-dimensional real normed linear space M ^d is reduced, if Δ(P) < Δ(R) for every convex body P ⊂ R different from R. The symbol Δ(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M ^d . We establish a number of properties of reduced bodies in M ². They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M ² is larger than Δ(R) for all directions strictly between two fixed directions and equals Δ(R) for these two directions.     

Convex polytopes whose projection bodies and difference sets are polars

In a paper by the author and B. Weissbach it was proved that the projection body and the difference set of ad-simplex (d≥2) are polars. Obviously, ford=2 a convex domain has this property if and only if its difference set is bounded by a so-called Radon curve. A natural question emerges about further classes of convex bodies inR ^d (d≥3) inducing the mentioned polarity. The aim of this paper is to show that a convexd-polytope (d≥3) is a simplex if and only if its projection body and its difference set are polars.     

Quadrangles Inscribed in a Closed Curve and the Vertices of a Curve

Let ABCDE be a pentagon inscribed in a circle. It is proved that if is a C⁴-generic smooth convex planar oval with four vertices (stationary points of curvature), then there are two similarities φ such that the quadrangle φ(ABCD) is inscribed in and the point φ(E)lies inside , as well as two similarities ψ such that the quadrangle ψ(ABCD) is inscribed in and ψ(E)lies outside . Itisalsoprovedthatif n is odd, then any smoothly embedded circle γ ↪ ℝⁿ contains the vertices of an equilateral (n + 1)-link polygonal line lying in a hyperplane of ℝⁿ. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 241–251.     

Several theorems of combinatorial geometry

A set is said to be H-convex if it can be represented by an intersection of a family of closed half-spaces whose outer normals belong to a given subset of the set H of the unit sphereS ⁿ⁻¹⊂R. On the basis of Helly’s theorem for H-convex sets recently obtained by us, we prove in this note certain extensions of Blaschke’s theorem (on the radius of an inscribed sphere) and of several other well-known theorems of combinatorial geometry. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 117–124, January, 1977.