Notes on L-/M-convex functions and the separation theorems

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis. Received: November 27, 1997 / Accepted: December 16, 1999?Published online May 12, 2000     

Multiplier Rules and Saddle-Point Theorems for Helbig’s Approximate Solutions in Convex Pareto Problems

This paper deals with approximate Pareto solutions in convex multiobjective optimization problems. We relate two approximate Pareto efficiency concepts: one is already classic and the other is due to Helbig. We obtain Fritz John and Kuhn–Tucker type necessary and sufficient conditions for Helbig’s approximate solutions. An application we deduce saddle-point theorems corresponding to these solutions for two vector-valued Lagrangian functions.     

The Central Limit Problem for Random Vectors with Symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries. This work was done while at Stanford University.     

A Production Lot-Size Model for Perishable Items Under Two Level Trade Credit Policy for a Retailer with a Powerful Position in a Supply Chain System

This paper investigates a production lot-size inventory model for perishable items under two levels of trade credit for a retailer to reflect the supply chain management situation. We assume that the retailer maintains a powerful position and can obtain full trade credit offered by supplier yet retailer just offers the partial trade credit to customers. Under these conditions, retailer can obtain the most benefits. Then, we investigate the retailer’s inventory policy as a cost minimization problem to determine the retailer’s inventory policy. A rigorous mathematical analysis is used to prove that the annual total variable cost for the retailer is convex, that is, unique and global-optimal solution exists. Mathematical theorems are developed to efficiently determine the optimal ordering policies for the retailer. The results in this paper generalize some already published results. Finally, numerical examples are given to illustrate the theorems and obtain a lot of managerial phenomena.     

Modifications of the Ricci tensor and applications

By using two modified Ricci tensors, we prove some theorems which correspond to Myers’s diameter estimate theorem and Bochner’s vanishing theorem.     

Duality and saddle-points for convex-like vector optimization problems on real linear spaces

Usually, finite dimensional linear spaces, locally convex topological linear spaces or normed spaces are the framework for vector and multiojective optimization problems. Likewise, several generalizations of convexity are used in order to obtain new results. In this paper we show several Lagrangian type duality theorems and saddle-points theorems. From these, we obtain some characterizations of several efficient solutions of vector optimization problems (VOP), such as weak and proper efficient solutions in Benson’s sense. These theorems are generalizations of preceding results in two ways. Firstly, because we consider real linear spaces without any particular topology, and secondly because we work with a recently appeared convexlike type of convexity. This new type, designated GVCL in this paper, is based on a new algebraic closure which we named vector closure. This research for the second author was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.     

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45     

Discrete convex analysis

A theory of “discrete convex analysis” is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1–4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by A. Frank, S. Fujishige, L. Lovász and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min-max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar’s conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature.     

Mann iteration of weak convergence theorems in Banach space

In this paper, by using Mann＇s iteration process we will establish several weak convergence theorems for approximating a fixed point of k-strictly pseudocontractive mappings with respect to p in p-uniformly convex Banach spaces. Our results answer partially the open question proposed by Marino and Xu, and extend Reich＇s theorem from nonexpansive mappings to k-strict pseudocontractive mappings.     

Inequalities for dual quermassintegrals of mixed intersection bodies

In this paper, we first introduce a new concept ofdual quermassintegral sum function of two star bodies and establish Minkowski’s type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov-Fenchel inequality and the Brunn-Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak’s mixed prosection bodies inequalities.     

Multiplicity of solutions for a class of Kirchhoff type problems

In this paper we apply the （variant） fountain theorems to study the symmetric nonlinear Kirch- hoff nonlocal problems. Under the Ambrosetti-Rabinowitz＇s 4-superlinearity condition, or no Ambrosetti- Rabinowitz＇s 4-superlinearity condition, we present two results of existence of infinitely many large energy solutions, respectively.     

Moment inequalities and central limit properties of isotropic convex bodies

The object of our investigations are isotropic convex bodies , centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies – specified by bounds on the second and fourth moments – is invariant under forming ‘expanded joinsrsquo;. Considering a body K as above as a probability space and taking , we define random variables on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions . We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with respect to -convergence and -convergence of the corresponding densities. Received: 21 March 2001 / in final form: 17 October 2001 / Published online: 4 April 2002     

Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating convex bodies of constant width to the family of “fat” spindle convex bodies. Also, this leads to the spherical analog of the well-known Blaschke–Lebesgue problem.     

Provably recursive functions of constructive and relatively constructive theories

In this paper we prove conservation theorems for theories of classical first-order arithmetic over their intuitionistic version. We also prove generalized conservation results for intuitionistic theories when certain weak forms of the principle of excluded middle are added to them. Members of two families of subsystems of Heyting arithmetic and Buss-Harnik’s theories of intuitionistic bounded arithmetic are the intuitionistic theories we consider. For the first group, we use a method described by Leivant based on the negative translation combined with a variant of Friedman’s translation. For the second group, we use Avigad’s forcing method.     

Tolerance in Helly-Type Theorems

In this paper we introduce the notion of tolerance in connection with Helly-type theorems and prove, using the Erdős–Gallai theorem, that any Helly-type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number of exceptional sets or points. In particular, we analyze some of the classical Helly-type theorems, such as Caratheodory’s and Tverberg’s theorems, as well as some other interesting ones.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

Discrete and Lexicographic Helly-Type Theorems

Helly’s theorem says that if every d+1 elements of a given finite set of convex objects in ℝ ^d have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems—where the common point should belong to an a-priori given set, lexicographic Helly theorems—where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We study the relations between the different types of the Helly theorems. We obtain several new discrete and lexicographic Helly numbers. An extended abstract containing parts of this work appeared in the proceedings of the Forty-Fifth Annual Symposium on Foundations of Computer Science (FOCS) 2004. This work is part of the author’s Ph.D. thesis, prepared in the school of mathematical sciences at Tel Aviv University, under the supervision of Professor Arie Tamir.     

Über einige Eindeutigkeitssätze für konvexe Körper

In this note,three unity theorems for compact convex bodies with interior points in euclidean space R_n up to translation resp. dilatation resp. (perspective) affinity are formulated.The first two of them are in a certain sense generalizations of well known theorems of K.VPOSS ans A.AEPPLI concerning convex hypersurfaces.All of them suppose relations of the supporting hyperplanes in corresponding points of two given convex bodies.Their proofs use the main theorem of calculus in the Lebesgue theory, and they have applications in the discussion of equality in some fundamental inequalities of the theory of convex bodies.     

Projections of convex bodies and the fourier transform

The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections ofl _p-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections ofl _p-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.     

Variational inclusions problems with applications to Ekeland’s variational principle,fixed point and optimization problems

In this paper, we prove the existence theorems of two types of systems of variational inclusions problem. From these existence results, we establish Ekeland’s variational principle on topological vector space, existence theorems of common fixed point, existence theorems for the semi-infinite problems, mathematical programs with fixed points and equilibrium constraints, and vector mathematical programs with variational inclusions constraints.