Conjugate points and closed geodesic arcs on convex surfaces

This paper discusses conjugate points on the geodesics of convex surfaces. It establishes their relationship with the cut locus. It shows the possibility of having many geodesics with conjugate points at very large distances from each other. It also shows that on many surfaces there are arbitrarily many closed geodesic arcs originating and ending at a common point. To achieve these goals, Baire category methods are employed.     

Typical convex curves on convex surfaces

In the sense of Baire categories, most convex curves on a smooth twodimensional closed convex surface are smooth. Moreover, if the set of all closed geodesics has empty interior in the space of all convex curves, then most convex curves are strictly convex.This paper was written during the author's visit at Western Washington University, whose substantial support is acknowledged.     

Generic Properties of Continuous Differential Inclusions and the Tonelli Method of Approximate Solutions

We study Cauchy problems for differential inclusions in Banach spaces and show that most such problems (in the sense of Baire’s categories) have solutions. We consider separately the cases where the point images of the right-hand side are compact and convex, and where they are merely bounded, closed and convex.     

Small sets in best approximation theory

The best approximation problem to a nonempty closed set in a locally uniformly convex Banach space is considered. The main result states that the set of points which have best approximation but the approximation problem is not well-posed is very small in a sense that it is σ-cone supported in the underlying space. This gives an improvement of an original result of Stečkin about the set of points with more than one best approximation which involves Baire categories. Examples on the necessity of some of the imposed conditions are provided.     

Densest packings of typical convex sets are not lattice-like

We show that ifP is a convex polygon which has no parallel sides, then the densest packing of the plane with congruent copies ofP is not lattice-like. As a corollary we obtain that, in the sense of Baire categories, for most convex disks densest packing is not lattice-like. This research was supported by the Hungarian National Foundation for Scientific Research (OTKA) under Grant Nos. 1907 and 14218.     

On typical degenerate convex surfaces

Various properties are given concerning geodesics on, and distance functions from points in, typical degenerate convex surfaces; i.e., surfaces obtained by gluing together two isometric copies of typical (in the sense of Baire category) convex bodies, by identifying the corresponding points of their boundaries.     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

The Determination of a Convex Set from Its Angle Function

In this paper we discuss the followingquestion: how can we decide whether a convex setis determined by its angle function or not? We givesufficient conditions for convex polygons and forregular convex sets which guarantee that the setis distinguishable. We also investigate the question: which setsare typical (in the sense of Baire category), thosewhich are distinguishable or those which are not?We prove that the family of distinguishable sets is ofsecond Baire category.     

On two conjectures of franz hering about convex surfaces

In this paper we show that from the point of view of the Baire categories for most convex bodies no shadow boundary is included in a hyperplane. A related, more quantitative question is also considered. It receives in general a negative answer and in theC ²-case a positive one, but remains open in theC ¹-case.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

向量拟平衡问题的本质解及解集的本质连通区

本文研究向量拟平衡问题,得到了向量拟平衡问题解的一个存在性结果,证明了在满足一定的连续性和凸性条件的问题构成的空间Y中,大多数(在Baire分类意义下)问题的解集是稳定的,并证明Y的某子集中,每个向量拟平衡问题的解集中至少存在一个本质连通区。作为应用,我们导出了多目标广义对策弱Pareto-Nash平衡点的存在性,证明了在满足一定的连续性和凸性条件的多目标广义对策构成的空间P中,大多数对策的弱Pareto-Nash平衡点是稳定的,并证明了P中的每个对策的弱Pareto-Nash平衡点集中至少有一个本质连通区。     

Local Baire classification of Lyapunov exponents

One considers two different definitions of the Baire class of a functional at a point. These definitions are in agreement with the common definition of the Baire class. The semicontinuity of a functional at a point is associated with its inclusion into the first Baire class at that point in the sense of the said definitions for Lyapunov exponents of a homogeneous nth-order system. In particular, it is shown that for the two smallest exponents, the inclusion into the first Baire class at a point is equivalent to semicontinuity in the sense of one of the two definitions and continuity in the sense of the other. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 56–70, 2007.     

The dimension print of most convex surfaces

The dimension print is a concept which contains more detailed information than the usual Hausdorff dimension. So, for example, a sphere and the surface of a cube have same dimension but different dimension prints. Can anything be said about the dimension print of most convex surfaces (in the Baire category sense)?     

The Generic Stability of Fixed Points for Upper Semicontinuous Mappings in Hyperconvex Metric Spaces

The main goal of this paper is to establish the generic stability of Fan-Glicksberg type fixed points in hyperconvex metric spaces. In order to do so, we first give Fan-Glicksberg type fixed point theorem in hyperconvex metric spaces and then the generic stability of fixed points for upper semicontinuous set-valued mappings is obtained. Our generic stability results show that almost all of fixed points of upper semicontinuous set-valued mappings defined in compact hyperconvex metric spaces are stable in the sense of Baire category theory     

Most continuous descent methods converge

We consider continuous descent methods for the minimization of convex functions defined on a general Banach space and show that most of them (in the sense of Baire category) converge.Received: 21 July 2004     

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

关于集值映射的随机不动点的稳定性

本文研究随机集值映射不动点的稳定性。通过集值分析,得到了随机集值不动点的本质稳定集的存在性。在Baire分类意义下,大多数的随机集值映射的随机不动点都是本质稳定的。这些推广了现有文献中的相应结果。     

Some subclasses of the real-valued honorary Baire two functions on ℝ n

Certain subclasses of the class of Baire one real-valued functions have very nice properties, especially concerning their points of continuity and their preservation of connectedness for many connected sets. A Gibson [weakly Gibson] is defined by the requirement that \(f(\overline{U})\subseteq\overline{f(U)}\) for every open [open connected] set U?? ⁿ . It is known that Baire one, Gibson functions are continuous, and that Baire one, weakly Gibson functions have Darboux-like properties in the sense that if U is an open connected set and \(U\subseteq S\subseteq\overline{U}\), then f(S) is an interval. Here we study the situation where the Baire one condition is replaced by honorary Baire two. Distinctly different results are found.     

A Plasticity Principle of Convex Quadrilaterals on a Convex Surface of Bounded Specific Curvature

We derive the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature and prove a plasticity principle which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point their endpoints form a convex quadrilateral and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight corresponding to a shortest arc causes a decrease of the two weights that correspond to the two neighboring shortest arcs and an increase of the weight corresponding to the opposite shortest arc by solving the inverse weighted Fermat-Torricelli problem for quadrilaterals on a convex surface of bounded specific curvature. The invariance of the weighted Fermat-Torricelli point(geometric plasticity principle) and the plasticity principle of quadrilaterals characterize the evolution of quadrilaterals on a complete convex surface. Furthermore, we show a connection between the plasticity of convex quadrilaterals on a complete convex surface with bounded specific curvature with the plasticity of some generalized convex quadrilaterals on a manifold which is certainly composed by triangles. We also study some cases of symmetrization of weighted convex quadrilaterals by introducing a new symmetrization technique which transforms some classes of weighted geodesic convex quadrilaterals on a convex surface to parallelograms in the tangent plane at the weighted Fermat-Torricelli point of the corresponding quadrilateral. This geometric method provides some pattern for the variable weights with respect to the 4-inverse weighted Fermat-Torricelli problem such that the weighted Fermat-Torricelli point remains invariant. By introducing the notion of superplasticity, we derive as an application of plasticity the connection between the Fermat-Torricelli point for some weighted kites with the fundamental equation of P. de Fermat for real exponents in the two dimensional Euclidean space. By using as an initial condition to the 3 body problem the solution of the 3-inverse weighted Fermat-Torricelli problem we give some future perspectives in plasticity, in order to derive new periodic solutions (chronotrees). We conclude with some philosophical ideas regarding Leibniz geometric monad in the sense of Euclid which use as an internal principle the plasticity of quadrilaterals.