On Concavity of the Monopolist's Problem Facing Consumers with Nonlinear Price Preferences

A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent x will choose to buy that product y(x) which maximizes his own utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this paper, we provide a necessary and sufficient condition for the convexity or concavity of the principal's (bilevel) optimization problem, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e., quasilinear) function of the price paid. Concavity, when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences. © 2019 Wiley Periodicals, Inc.     

On a generalization of Blaschke's Rolling Theorem and the smoothing of surfaces

A generalization of Blaschke's Rolling Theorem for not necessarily convex sets is proved that exhibits an intimate connection between a generalized notion of convexity, various concepts in mathematical morphology and image processing, and a certain smoothness condition. As a consequence a geometric characterization of Serra's regular model is obtained and various problems in image processing arisng from the smoothing of surfaces with Sternberg's rolling ball algorithm are addressed. Copyright © 1999 John Wiley & Sons, Ltd.     

Convex upper and lower bounds for present value functions

In this paper we present an efficient methodology for approximating the distribution function of the net present value of a series of cash‐flows, when discounting is presented by a stochastic differential equation as in the Vasicek model and in the Ho–Lee model. Upper and lower bounds in convexity order are obtained. The high accuracy of the method is illustrated for cash‐flows for which no analytical results are available. Copyright © 2001 John Wiley & Sons, Ltd.     

Fej\'{e}r type inequalities for Harmonically-convex functions with applications

In this paper, a new weighted identity involving harmonically symmetric functions and differentiable functions is established. By using the notion of harmonic symmetricity, harmonic convexity and some auxiliary results, some new Fej\''{e}r type integral inequalities are presented. Applications to special means of positive real numbers are given as well.     

Kostant's convexity theorem and the compact classical groups

The relationship between the classical Schur-Horn's theorem on the diagonal elements of a Hermitian matrix with prescribed eigenvalues and Kostant's convexity theorem in the context of Lie groups. By using Kostant's convexity theorem, we work out the statements on the special orthogonal group and the symplectic group explicitly. Schur-Horn's result can be stated in terms of a set of inequalities. The counterpart in the Lie-theoretic context is related to a partial ordering, introduced by Atiyah and Bott, defined on the closed fundamental Weyl chamber. Some results of Thompson on the diagonal elements of a matrix with prescribed singular values are recovered. Thompson-Poon's theorem on the convex hull of Hermitian matrices with prescribed eigenvalues is also generalized. Then a result of Atiyah-Bott is recovered.     

Singular Abreu Equations and Minimizers of Convex Functionals with a Convexity Constraint

We study the solvability of second boundary value problems of fourth-order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the monopolist's problem in economics. The right-hand sides of our Abreu-type equations are quasilinear expressions of second order; they are highly singular and a priori just measures. However, our analysis in particular shows that minimizers of the 2D Rochet-Choné model perturbed by a strictly convex lower-order term, under a convexity constraint, can be approximated in the uniform norm by solutions of the second boundary value problems of singular Abreu equations. © 2019 Wiley Periodicals, Inc.     

A generalized section theorem and a minimax inequality for a vector-valued mapping 1

A generalized Fan's section theorem has proposed by replacing convexity assumptions with merely topological properties. A generalized reformulation of Browder's fixed point theorem has derived. The Minimax Inequalities for vector-valued mapping in an ordered Banach space have established without the convexity and with convexity, respectively.     

Abstract Convexity,Some Relations and Applications

The aim of this article is to analyze the relationship between various notions of abstract convexity structures that we find in the literature, in connection with the problem of the existence of continuous selections and fixed points of correspondences. We focus mainly on the notion of mc -spaces, which was introduced in [J.V. LLinares (1998). Unified treatment of the problem of the existence of maximal elements in binary relations: a characterization. Journal of Mathematical Economics , 29 , 285-302], and its relationship with c -spaces [Ch.D. Horvath (1991). Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications , 156 , 341-357], simplicial convexity [R. Bielawski (1987). Simplicial convexity and its applications. Journal of Mathematical Analysis and Applications , 127 , 155-171], order convexity (used in [Ch.D. Horvath and J.V. LLinares (1996). Maximal elements and fixed points for binary relations on topological ordered spaces. Journal of Mathematical Economics , 25 , 291-306]), B '-simplicial convexity and L -spaces [H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano and J.V. LLinares (1998). Abstract convexity and fixed points. Journal of Mathematical Analysis and Applications , 222 , 138-150]. Moreover, in the context of mc -spaces, a characterization result of nonempty finite intersection, in the line with the Knaster-Kuratowski-Mazurkiewicz Lemma, some consequences of it and some generalizations of Browder's existence of continuous selection and fixed point theorem are presented.     

Necessary and Sufficient Convexity Conditions for the Ranges of Vector Measures

The absence of atoms in Lyapunov's Convexity Theorem is a sufficient, but not a necessary condition for the convexity of the range of an n - dimensional vector measure. In this paper algebraic and topological convexity conditions generalizing Lyapunov's Theorem are developed which are sufficient and necessary as well. From these results the converse of Lyapunov's Theorem is derived in the form of a nonconvexity statement which gives insight into the geometric structure of the ranges of vector measures with atoms. Further, a characterization of the one-dimensional faces of a zonoid Z_μ, is given with respect to the generating spherical Borel measure μ. As an application, it is shown that the absence of μ - atoms is a necessary and sufficient convexity condition for the range of the indefinite integral ∫ x dμ where x denotes the identical function on S^n-1.     

One class of separable optimization problems: solution method,application

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Craven's notion of K-invexity function (when K is a cone in ? ⁿ ) and Martin's notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.     

On Eckhoff's conjecture for Radon numbers; or how far the proof is still away

Eckhoff's conjecture for the Τ-Radon numbers r(Τ) of a convexity space. (X,C) says r(Τ) ≦ (r?1)(Τ?1)+1, with r = r(2). The main result of this note is that Eckhoff's conjecture is true in case ¦X¦ ≦ 2r and Τ = 3, i.e. each (2r?1)-set in a space with 2r?1 or 2r elements has a 3-Radon partition.     

张量积上参数Bezier曲面保凸的充分条件

张量积上参数Ｂｅｚｉｅｒ曲面保凸的充分条件朱功勤，殷明（合肥工业大学）ＴＨＥＣＯＮＶＥＸＩＴＹＯＦＰＡＲＡＭＥＴＲＩＣＢＥＺＩＥＲＰＡＴＣＨＯＶＥＲＲＥＣＴＡＮＧＬＥＳ￥ＺｈｕＧｏｎｇ－ｑｉｎ；ＹｉｎＭｉｎｇ（ＨｅｆｅｉＵｎｉｖｅｒｓｉｔｙ）Ａｂｓｔ...     

具有特殊坐标曲线网的B\'ezier曲面

在计算机辅助几何设计中, B\''ezier曲面是一类重要的参数曲面.在微分几何中,坐标曲线网也是重要的研究内容.本文中,我们对具有特殊坐标曲线网(如正交曲线网、曲率曲线网、共轭曲线网等)的B\''ezier曲面进行研究.此外,我们还构造了满足能量约束的特殊B\''ezier曲面,给出了基于控制结构的充分条件并给出具体实例.     

Essential components of the set of weakly Pareto-Nash equilibrium points

In this paper, we first give a generalization of Ky Fan's inequality to vector-valued functions. We prove that, for every vector-valued function (satisfying some continuity and convexity conditions), there exists at least one essential component of the set of its Ky Fan's points. As applications, we show that, for every multiobjective game (satisfying some continuity and convexity conditions), there exists at least one essential component of the set of its weakly Pareto-Nash equilibrium points.     

Study of the Eddy-Current Problem Taking Account of Hall's Effects

AMS(MOS): 35K

The parabolic problem corresponding to Maxwell's equations in a metal conductor is studied taking account of Hall's effect,i.e. the dependence of resistivity of the magnetic field. A variational formulation is given and an existence theorem is proved by using a compactness result based on a strict convexity property.     

Even and odd marginal worth vectors,Owen's multilinear extension and convex games

In this paper we characterize convex games by means of Owen's multilinear extension and the marginal worth vectors associated with even or odd permutations. Therefore we have obtained a refinement of the classic theorem; Shapley (1971), Ichiishi (1981) in order to characterize the convexity of a game by its marginal worth vectors. We also give new expressions for the marginal worth vectors in relation to unanimity coordinates and the first partial derivatives of Owen's multilinear extension. A sufficient condition for the convexity is given and also one application to the integer part of a convex game.     

Convexity estimates for the solutions of a class of semi-linear elliptic equations

In this paper, we are concerned with convexity estimates for solutions of a class of semi-linear elliptic equations involving the Laplacian with power-type nonlinearities. We consider auxiliary curvature functions which attain their minimum values on the boundary and then establish lower bound convexity estimates for the solutions. Then we give two applications of these convexity estimates. We use the deformation method to prove a theorem concerning the strictly power concavity properties of the smooth solutions to these semi-linear elliptic equations. Finally, we give a sharp lower bound estimate of the Gaussian curvature for the solution surface of some specific equation by the curvatures of the domain's boundary.     

On Kostant's partial order on hyperbolic elements

We study Kostant's partial order on the elements of a semisimple Lie group in relations with the finite-dimensional representations. In particular, we prove the converse statement of Theorem 6.1 given by Kostant [B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. 6 (1973), pp. 413–455] on hyperbolic elements.     

On a very weak bernoulli condition †

E. Eberlein has proposed a generalization of D. S. Ornstein's very weak Bernoulli condition for strictly stationary sequences of r.v.'s; Eberlein's generalization is motivated by probability-theoretic (rather than ergodic-theoretic) questions. Dehling, Denker and Philipp showed that the mixing rate in this definition of vwB cannot be o(l/n) except when (X?k$esup;) is i.i.d. Here we construct a class of strictly stationary sequences, and with this class we show that, in essence, any mixing rate that satisfies a mild convexity condition and is not o(1/n) is possible for this vwB condition, and is compatible with a certain condition on maximal correlations which implies nice moment properties useful in proving limit theorems.     

Poisson Lie Groups and Non-Linear Convexity Theorems

In this paper we describe how one can derive nonlinear convexity theorems which are similar to Kostant's nonlinear convexity theorem for the Iwasawa decomposition from Hamiltonian actions on Poisson Lie groups. Our results generalize those of LU and Ratiu ([24]) and aim at a unified symplectic framework for the convexity theorem in [32] and the linear convexity theorems for coadjoint orbits of convex type in [15].