Compactification of Strictly Convex Real Projective Structures

In this paper we compactify the space of convex real projective structures, known as Hitchin’s component in a representation variety. We give geometric meanings to the boundary points.Mathematics Subject Classifications (2000). 51M10, 57S25.     

Computing Proximal Points of Convex Functions with Inexact Subgradients

Locating proximal points is a component of numerous minimization algorithms. This work focuses on developing a method to find the proximal point of a convex function at a point, given an inexact oracle. Our method assumes that exact function values are at hand, but exact subgradients are either not available or not useful. We use approximate subgradients to build a model of the objective function, and prove that the method converges to the true prox-point within acceptable tolerance. The subgradient g _k used at each step k is such that the distance from g _k to the true subdifferential of the objective function at the current iteration point is bounded by some fixed ε > 0. The algorithm includes a novel tilt-correct step applied to the approximate subgradient.     

The Classification of Symplectic Structures of Convex Type

In this paper we give a classification of a certain class of semisimple symplectic structures, more precisely all symplectic structures for which a symplectic module (V,) is of convex type. This classification then leads to a classification of Lie algebras with invariant cones and at most one dimensional center.     

Computing the Largest C-Eigenvalue of a Tensor Using Convex Relaxation

Journal of Optimization Theory and Applications - A piezoelectric-type tensor is of order three that is symmetric with respect to the last two indices. Its largest C-eigenvalue determines the...     

Computing Skeletons for Rectilinearly Convex Obstacles in the Rectilinear Plane

Journal of Optimization Theory and Applications - We introduce the concept of an obstacle skeleton, which is a set of line segments inside a polygonal obstacle $$omega $$ that can be used in place...     

Computing an Integer Point of a Class of Convex Sets

A simplicial algorithm is proposed for computing an integer point of a convex set ^CRn satisfying

具凸结构的概率度量空间中的重合点定理

本文在具凸结构的概率度量空间中，对非线性混合压缩映象得出了几个重合点和公共不动点定理。     

An Algorithm for Computing the Best Convex Contrast Curve of Two Bivariate Samples

Abstract

Let (x ₁, y ₁),…, (x _n, y _n) be a sample of points in , consisting of two subsamples (x ₁, y ₁),…, (x _n1, y _n1) and (x _n1+1, y _n1+1),…, (x _n, y _n), where n = n ₁ + n ₂. We consider the problem of separating the two subsamples by a convex contrast curve k…that is, a real-valued convex function x → k(x). For given curve k we consider the relative frequency p ₁ of points of the first sample above the graph of k similarly p ₂. The goal is to choose k such that the difference in relative frequency p ₁ – p ₂ becomes maximal. This maximal value can be regarded as a generalized one-sided Kolmogorov-Smirnov test statistic , where p ₁(A) = (1/n ₁)#{i ≤ n ₁ : (xi, y _i) ∈ A}, p ₂(A) = (1/n ₂#{n ₁ + 1 ≤ i ≤ n : (x _i, y _i) ∈ A} and is the class of all epigraphs {(x y) : k(x) ≤ y} of convex functions k. The standard Kolmogorov-Smirnov test statistic corresponds to the class consisting of all sets —that is, epigraphs of constant functions. Test statistics of this form arise in nonparametric analysis of covariance where the regression function is assumed to be convex. In this article we give a recursive relation and an algorithm based on it to compute the value KS and an optimal convex contrast curve in O(n ³) steps. The convexity assumptions on the sets of and on the nonparametric regression function are interrelated; this is explained in the large sample situation where n ₁ Λ n ₂ → + ∞.     

结构的失效可能度及模糊概率计算方法

依据模糊可能性理论,系统地建立含模糊变量时结构的可靠性计算模型。旨在解决模糊结构、模糊-随机结构和模糊状态假设下结构的可靠性计算问题。所建模型可给出模糊结构失效的可能度和模糊-随机结构失效概率的可能性分布。研究表明:对同时含模糊变量和随机变量的混合可靠性计算问题,把失效概率(或可靠度)作为模糊变量,能更客观地反映系统的安全状况。算例分析说明了文中方法的合理性和有效性。     

Finding Convex Sets in Convex Position

Let F{\cal{F}} denote a family of pairwise disjoint convex sets in the plane. F{\cal{F}} is said to be in convex position, if none of its members is contained in the convex hull of the union of the others. For any fixed k 3 5k\ge5, we give a linear upper bound on P_k(n)P_k(n), the maximum size of a family F{\cal{F}} with the property that any k members of F{\cal{F}} are in convex position, but no n are.     

Convex Defining Functions for Convex Domains

We give three proofs of the fact that a smoothly bounded, convex domain in ℝ ⁿ has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.     

凸体间的逼近

For the affine distance d(C, D) between two convex bodies C, D C R^n, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) ≤ n^1/2 if one is an ellipsoid and another is symmetric, d(C, D) ≤ n if both are symmetric, and from F. John's result and d(C1, C2) ≤ d(C1, C3)d(C2, C3) one has d(C, D) ≤ n^2 for general convex bodies; M. Lassak proved d(C, D) ≤ (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asvmmetrv for convex bodies.     

Convex Dimension of Locally Planar Convex Geometries

We prove that convex geometries of convex dimension n that satisfy two properties satisfied by nondegenerate sets of points in the plane, may have no more than 2 ^n-1 points. We give examples of such convex geometries that have n \choose 4 + n \choose 2 + n \choose 0 points. Received June 7, 1999, and in revised form April 18, 2000. Online publication September 22, 2000.     

On Convex Class of Pairs of Convex Bodies

In this paper we introduce a quotient class of pairs of convex bodies in which every member have convex union.

     

K-Drop凸空间与局部K-Drop凸空间

引入了Banach空间的局部k-drop凸性质，研究了k-drop凸与局部k-drop凸的一些性质以及两者之间的关系，并用单位球的切片统一而简洁地处理了这两个性质．     

Convex billiards

This article deals with (partly rather peculiar) properties of typical convex billiards. In particular, convex caustics, trajectories which terminate on the boundary and dense trajectories are investigated.Dedicated to Professor Curt Christian on the occasion of his 70th birthday     

Minimal Convex k-Gons Containing a Given Convex Polygon

There is a k-gon of minimal area containing a given convex n-gon (k<n) such that k-1 sides of the n-gon lie on the sides of the k-gon. All midpoints of the sides of the k-gon belong to the n-gon. Bibliography: 3 titles.     

Maximization of Generalized Convex Functionals in Locally Convex Spaces

The major part of the investigation is related to the problem of maximizing an upper semicontinuous quasiconvex functional f over a compact (possibly nonconvex) subset K of a real Hausdorff locally convex space E. A theorem by Bereanu (Ref. 1) says that the condition f is quasiconvex (quasiconcave) on K is sufficient for the existence of maximum (minimum) point of f over K among the extreme points of K. But, as we prove by a counterexample, this is not true in general. On the further condition that the convex hull of the set of extreme points of K is closed, we show that it is sufficient to claim that f is induced-quasiconvex on K to achieve an equivalent conclusion. This new concept of quasiconvexity, which we define by requiring that each lower-level set of f can be represented as the intersection of K with some convex set, is suitable for functionals with a nonconvex domain. Under essentially the same conditions, we prove that an induced-quasiconvex functional f is directionally monotone in the sense that, for each y K, the functional f is increasing along a line segment starting at y and running to some extreme point of K. In order to guarantee the existence of maximum points on the relative boundary r K of K, it suffices to make weaker demands on the function f and the space E. By introducing a weaker kind of directional monotonicity, we are able to obtain the following result: If f is i.s.d.-increasing i.e., for each y y K, there is a half-line emanating from y such that f is increasing along this half-line, then f attains its maximum at rK , even if E is a topological linear Hausdorff space (infinite-dimensional and not necessarily locally convex). We state further a practical method of proving i.s.d.-monotonicity for functions in finite-dimensional spaces and we discuss also some aspects of classification.