An Unsolved Problem of Fenchel

The Fenchel problem of level sets is solved under the conditions that theboundaries of the nested family of convex sets in R^n>+1 aregiven by C³ n-dimensional differentiable manifolds and theconvex sets determine an open or closed convex set inRⁿ⁺¹.     

Fenchel’s Duality Theorem for Nearly Convex Functions

We present an extension of Fenchel’s duality theorem by weakening the convexity assumptions to near convexity. These weak hypotheses are automatically fulfilled in the convex case. Moreover, we show by a counterexample that a further extension to closely convex functions is not possible under these hypotheses. The authors are grateful to the Associate Editor for helpful suggestions and remarks which improved the quality of the paper. The second author was supported by DFG (German Research Foundation), project WA 922/1.     

Dual Problems and Separation of Sets: Lagrange and Fenchel Duality

An extremum problem is embedded in a parametric scheme which contains, as a particular case, the classic perturbation function. The introduction of the image of the embedded problem allows one to derive a generalized duality and, in particular, Lagrangian and Fenchel duality.     

A Fenchel Duality Aspect of Iterative I-Projection Procedures

In this paper we interpret Dykstra's iterative procedure for finding anI-projection onto the intersection of closed, convex sets in terms of itsFenchel dual. Seen in terms of its dual formulation, Dykstra's algorithm isintuitive and can be shown to converge monotonically to the correctsolution. Moreover, we show that it is possible to sharply bound thelocation of the constrained optimal solution.     

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called directed sets is constructed, in which the convex cone of all nonempty convex compact sets in R ⁿ is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in R ⁿ are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a support function and directed supporting faces of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the support function and recursively on the directed supporting faces. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.     

Convergence of Polynomial Level Sets

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.

     

Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations

We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.     

Level Function Method for Quasiconvex Programming

In this paper, we introduce the notion of level function for a continuous real-valued quasiconvex function. The existence, construction, and application of level functions are discussed. Further, we propose a numerical method based on level functions for the solution of quasiconvex minimization problems. Several versions of the algorithms are presented. Also, we apply the idea of the level function method to the solution of a class of variational inequality problems. Finally, the results of numerical experiments on the proposed algorithms are reported.     

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets

This paper is a continuation of the author's first paper (Set-Valued Anal. 9 (2001), pp. 217–245), where the normed and partially ordered vector space of directed sets is constructed and the cone of all nonempty convex compact sets in R ⁿ is embedded. A visualization of directed sets and of differences of convex compact sets is presented and its geometrical components and properties are studied. The three components of the visualization are compared with other known differences of convex compact sets.     

Level Sets of Multiparameter Stable Processes

We establish the correct Hausdorff measure function for the level sets of additive strictly stable processes derived from strictly stable processes satisfying Taylor’s condition (A). This leads naturally to a characterization of local time in terms of the corresponding Hausdorff measure function of the level set.      

On a Problem of Fenchel

We give a necessary and sufficient condition for a given set of positive real numbers to be the dihedral angles of a hyperbolic n -simplex in this note. This answers a question of W. Fenchel raised in his book, Elementary Geometry in Hyperbolic Space, (De Gruyter, Berlin, 1989, p. 174) where he obtained some necessary conditions for which six numbers have to satisfy in order to be the dihedral angles of a hyperbolic tetrahedron. We also present a simple proof of the known necessary and sufficient condition for the dihedral angles of Euclidean n-simplexes.     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

On the Minimizing Trajectory of Convex Functions with Unbounded Level Sets

We consider a convex function f(x) with unbounded level sets. Many algorithms, if applied to this class of functions, do not guarantee convergence to the global infimum. Our approach to this problem leads to a derivation of the equation of a parametrized curve x(t), such that an infimum of f(x) along this curve is equal to the global infimum of the function on ⁿ .We also investigate properties of the vectors of recession, showing in particular how to determine a cone of recession of the convex function. This allows us to determine a vector of recession required to construct the minimizing trajectory.     

Sets with External Chebyshev Layer

Mathematical Notes -     

Conditions for Error Bounds and Bounded Level Sets of Some Merit Functions for the Second-Order Cone Complementarity Problem

Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform P ^*-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called R ₀₁ or R ₀₂-functions to keep the property of bounded level sets. This work is partially supported by National Science Council of Taiwan.     

A generalization of Fenchel duality theory

This paper deals with an extention of Fenchel duality theory to fractional extremum problems, i.e., problems having a fractional objective function. The main result is obtained by regarding the classic Fenchel theorem as a decomposition property for the extremum of a sum of functions into a sum of extrema of functions, and then by extending it to the case where the addition is replaced by the quotient. This leads to a generalization of the classic concept of conjugate function. Several remarks are made about the conceivable further generalizations to other kinds of decomposition.     

Finite Sets as Complements of Finite Unions of Convex Sets

Suppose S?? ^d is a set of (finite) cardinality n, whose complement can be written as the union of k convex sets. It is perhaps intuitively appealing that when n is large k must also be large. This is true, as is shown here. First the case in which the convex sets must also be open is considered, and in this case a family of examples yields an upper bound, while a simple application of a theorem of Björner and Kalai yields a lower bound. Much cruder estimates are then obtained when the openness restriction is dropped. For a given set S the problem of determining the smallest number of convex sets whose union is ? ^d ?S is shown to be equivalent to the problem of finding the chromatic number of a certain (infinite) hypergraph ? _S . We consider the graph \(\mathcal {G}_{S}\) whose edges are the 2-element edges of ? _S , and we show that, when d=2, for any sufficiently large set S, the chromatic number of \(\mathcal{G}_{S}\) will be large, even though there exist arbitrarily large finite sets S for which \(\mathcal{G}_{S}\) does not contain large cliques.     

Comparative Properties of Three Metrics in the Space of Compact Convex Sets

Along with the Hausdorff metric, we consider two other metrics on the space of convex sets, namely, the metric induced by the Demyanov difference of convex sets and the Bartels–Pallaschke metric. We describe the hierarchy of these three metrics and of the corresponding norms in the space of differences of sublinear functions. The completeness of corresponding metric spaces is demonstrated. Conditions of differentiability of convex-valued maps of one variable with respect to these metrics are proved for some special cases. Applications to the theory of convex fuzzy sets are given.