Extremal dilations and their associated extremal measures

If μ and λ are probability measures on a metrisable compact convex set with μ < λ in the Choquet sense, then the main object of this paper is the study of the extremal structure of the convex set of all dilations carrying μ to λ. The extremal dilations are characterized and the relationships between these dilations and the extremal measures they induce are investigated. Several examples of extremal dilations with special properties are given to illustrate their behavior. Also given is a systematic characterization of measures which are extreme in the convex set of all measures dominating μ in the Choquet ordering.     

Exterior Monge-Ampère solutions

We discuss the Siciak-Zaharjuta extremal function of a real convex body in Cn, a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R².     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Three dimensional line stochastic matrices and extreme points

Several properties of the extreme points of the convex set of three dimensional line stochastic matrices of order n are presented. The existence of many different classes of extremal configurations is established. These extremal matrices exhibit a large variety of patterns with some unexpected configurations. Latin squares of special types are used in some of the existence results. Furthermore, three questions raised by Brualdi and Csima are answered concerning the extreme points of three dimensional plane stochastic matrices of order n.     

Convex hull realizations of the multiplihedra

We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to An-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_∞-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph-associahedra.     

A two-dimensional inequality and uniformly continuous retractions

We prove a new inequality valid in any two-dimensional normed space. As an application, it is shown that the identity mapping on the unit ball of an infinite-dimensional uniformly convex Banach space is the mean of n uniformly continuous retractions from the unit ball onto the unit sphere, for every n?3. This last result allows us to study the extremal structure of uniformly continuous function spaces valued in an infinite-dimensional uniformly convex Banach space.     

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.     

Mixed Polytopes

Abstract. Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes simplices.     

Extremal properties of nondifferentiable convex functions on euclidean sets of combinations with repetitions

A general approach is suggested for studying extremal properties of nondifferentiable convex functions on Euclidean combinatorial sets. On the basis of this approach, by solving the linear optimization problem on a set of combinations with repetitions, we obtain estimates of minimum values of convex and strongly convex objective functions in optimization problems on sets of combinations with repetitions and establish sufficient conditions for the existence of the corresponding minima. Kharkov Institute of Radioelectronics, Kharkov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 680–691, June, 1994.     

Mixed Polytopes

   Abstract. Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes simplices.     

Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u^? for the semilinear elliptic equation −△u+c(x)⋅∇u=λf(u) on a bounded smooth domain of Rn with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a∈(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.     

Sur la fonction extrémale plurisousharmonique relative à deux convexes

We prove results which are parallel to Lempert's theorem on the plurisbharmonic Green function of a convex domain for the extremal plurisubharmonic function relative to two convex domains. Namely, its sublevel sets are convex and the region between the two domains is foliated by complex curves on which the relative extremal function is harmonic.     

Duality and optimality for convex extremal problems described by discrete inclusions

This paper is devoted to the Hamiltonian approach for extremal problems concerning convex (multi-valued) mapping. The approach exploits the concept of a Hamiltonian function permitting simplified proofs and useful mathematical insights. Moreover it provides in a duality framework a common point ox view upon the methods used. by Rockafellar, (the theory of convex processes), Pshenichnyi (the conjugate transformation method) and CASS (the symmetric duality scheme) to construct optimality conditions. The theory is used to develop a complete characterization of optimal solutions for multi-period convex programming problems.     

On a normed version of a Rogers–Shephard type problem

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes–Thompson, and Gromov’s mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.     

一些函数族的一般极值问题

本文主要讨论了带限制条件的正实部解析函数族及纯凸像函数族的一般极值问题．首先我们得了两类带限制条件的正实部函数族的支撑点的表达式．其次，我们讨论了亚纯凸像函数族的极值问题，得到了亚纯凸像函数族上Ｆｒｃｈｅｔ可导泛函所对应的极函数的最好形式．     

On support points for some families of univalent functions

Given a closed subset of the familyS* (α) of functions starlike of order α, a continuous Fréchet differentiable functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α). The support points ofS* (α) is completely characterized and shown to coincide with the extreme points of its convex hulls. Given any finite collection of support points ofS* (α), a continuous linear functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α).     

On estimates of minima of criterion functions in optimization on combinations

On the basis of the approach proposed, we obtain new estimates of extremal values of strongly convex differentiable functions and strengthened estimates of minima on a set of combinations with repetions. Poltava Technical University, Poltava. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 118–1121, August, 1999.     

Complex geometry of polygonal linkages

We present observations on the complex geometry of polygonal linkages arising in the framework of our approach to extremal problems on configuration spaces. Along with a few general remarks on applications of complex geometry and theory of residues, we present new results obtained in this way. Most of the new results are presented in the case of a planar quadrilateral linkage with generic lengths of the sides. First, we show that, for each configuration of planar quadrilateral linkage Q(a, b, c, d) with pairwise distinct side-lengths (a, b, c, d), the cross-ratio of its vertices belongs to the circle of radius ac/bd centered at the point $ 1\in \mathbb{C} $ . Next, we establish an analog of the Poncelet porism for a discrete dynamical system on a planar moduli space of a 4-bar linkage defined by the product of diagonal involutions and discuss some related issues suggested by a beautiful link to the theory of discrete integrable systems discovered by J. Duistermaat. We also present geometric results concerned with the electrostatic energy of point charges placed at the vertices of a quadrilateral linkage. In particular, we establish that all convex shapes of a quadrilateral linkage arise as the global minima of a system of charges placed at its vertices, and these shapes can be completely controlled by the value of the charge at just one vertex, which suggests a number of interesting problems. In conclusion, we describe a natural connection between certain extremal problems for configurations of linkage and convex polyhedra obtained from its configurations using the Minkowski 1897 theorem and present a few related remarks.     

Constrained best approximation in Hilbert space

In this paper we study the characterization of the solution to the extremal problem inf{‖x‖x ∈C ∩M}, wherex is in a Hilbert spaceH, C is a convex cone, andM is a translate of a subspace ofH determined by interpolation conditions. We introduce a simple geometric property called the “conical hull intersection property” that provides a unifying framework for most of the basic results in the subject of optimal constrained approximation. Our approach naturally lends itself to considering the data cone as opposed to the constraint cone. A nice characterization of the solution occurs, for example, if the data vector associated withM is an interior point of the data cone.     

Minimization of a convex functional in a linear system of delay differential equations with fixed ends

A numerical method is proposed for solving the problem of moving a dynamic object described by a system of linear differential-difference equations to the origin with the minimization of a nonnegative convex functional. The method is proved to converge globally to an ?-optimal solution. The ?-optimal solution is understood as an extremal control u(t), t ∈ [0, T], that moves the system to the ?-neighborhood of the origin.