Derivatives of generalized farthest functions and existence of generalized farthest points

The relationship between directional derivatives of generalized farthest functions and the existence of generalized farthest points in Banach spaces is investigated. It is proved that the generalized farthest function generated by a bounded closed set having a one-sided directional derivative equal to 1 or −1 implies the existence of generalized farthest points. New characterization theorems of (compact) locally uniformly convex sets are given.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

Characterizations of simultaneous farthest point in normed linear spaces with applications

In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space X. We study simultaneous farthest point in X by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball of X ^*, where X ^* is the dual space of X. We establish the characterizations of simultaneous farthest points for bounded sets in , the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm . It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [9, Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in X which specifically concentrates on the Hahn-Banach Theorem in X.     

Existence of fixed points for some convex operators and applications to multi-point boundary value problems

Existence results of fixed points for some convex operators are given by means of fixed point theorem of cone expansion and compression, then they are applied to nonlinear multi-point boundary value problems.     

Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces

In this paper, we consider an implicit iteration process to approximate the common fixed points of two finite families of asymptotically quasi-nonexpansive mappings in convex metric spaces. As a consequence of our result, we obtain some related convergence theorems. Our results generalize some recent results of Khan and Ahmed [4], Khan et al. [6], Sun [12], Wittmann [14] and Xu and Ori [15].     

The AR-property for Roberts' example of a compact convex set with no extreme points Part 1: general result

We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR.

     

Continuity at the extreme points

We prove that a bounded convex lower semicontinuous function defined on a convex compact set K is continuous at a dense subset of extreme points. If there is a bounded strictly convex lower semicontinuous function on K, then the set of extreme points contains a dense completely metrizable subset.     

The Hermite-Hadamard inequality for convex functions on a global NPC space

We prove an extension of Choquet's theorem to the framework of compact metric spaces with a global nonpositive curvature. Together with Sturm's extension [K.T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: Pascal Auscher, et al. (Eds.), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs April 16-July 13, 2002, Paris, France, in: Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357-390] of Jensen's inequality, this provides a full analogue of the Hermite-Hadamard inequality for the convex functions defined on such spaces.     

The AR-property for Roberts' example of a compact convex set with no extreme points Part 2: Application to the example

In this second part of our paper, we apply the result of Part 1 to show that the compact convex set with no extreme points, constructed by Roberts (1977), is an AR.

     

Limit theorems for the convex hull of random points in higher dimensions

We give a central limit theorem for the number of vertices of the convex hull of independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in that includes the normal family. Furthermore, we prove that, among these distributions, the variance of exhibits the same order of magnitude as the expectation as The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.

     

On the lattice points on strictly convex surfaces

The aim of this paper is to show that on strictly convex closed surface in the three-dimensional space there cannot be more than $\text{(2.2 + o(1))P}{}^{\mathrm{<mtext>3</mtext><mtext>4</mtext>}}$ points with integral coordinates, where P is the surface area.     

A convergence analysis for a convex version of Dikin's algorithm

This paper is concerned with the convergence property of Dikin's algorithm applied to linearly constrained smooth convex programs. We study a version of Dikin's algorithm in which a second-order approximation of the objective function is minimized at each iteration together with an affine transformation of the variables. We prove that the sequence generated by the algorithm globally converges to a limit point at a local linear rate if the objective function satisfies a Hessian similarity condition. The result is of a theoretical nature in the sense that in order to ensure that the limit point is an -optimal solution, one may have to restrict the steplength to the order ofO(). The analysis does not depend on non-degeneracy assumptions.     

Covariance identities for normal variables via convex polytopes

Siegel (1993) presented a covariance identity involving normal variables that seems to flout notions of dependence. Here we show that it has an explanation from an unexpected quarter: convex geometry and the centroid known as the Steiner point.     

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.     

Extreme points of the distance function on convex surfaces

We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that ``almost all" points admit a single farthest point on the surface.

     

A decomposition method for convex minimization problems and its application

1. IntroductionConsider the following special convex programming problem(P) adn{f(~) g(z); Ax = z},where f: Re - (--co, co] and g: Re - (--co, co] are closed proper convex functions andA is an m x n matrix. The Lagrangian for problem (P) is defined by L: Rad x Re x Re -- (~co, co] as follows:L(x, z, y) = f(x) g(z) (y, Ax ~ z), (1.1)where (., .) denotes the inner product in the general sense and 'y is the Lagrangian multiplierassociated with the constraint Ax = z. The augmented L…     

Some results on the convex hull of finitely many convex sets

A better than quadratic estimate is given for the volume of the convex hull of points on Hadamard manifolds with pinched curvature. It was known previously that the volume is bounded by some polynomial in . The estimate comes from the study of the convex hull of finitely many convex sets on Hadamard manifolds.

     

Estimates for the number of rational points on convex curves and surfaces

Let Γ ⊂ ℝ^d be a bounded strictly convex surface. We prove that the number k_n(Γ) of points of Γ that lie on the lattice satisfies the following estimates: lim inf k_n(Γ)/n^d−2 < ∞ for d ≥ 3 and lim inf k_n(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 174–189.     

Farthest points and cut loci on some degenerate convex surfaces

We study farthest points and cut loci on doubly covered convex polygons, and determine them explicitly on doubly covered n-dimensional simplices.