Orlicz projection bodies

Minkowski's projection bodies have evolved into Lp projection bodies and their asymmetric analogs. These all turn out to be part of a far larger class of Orlicz projection bodies. The analog of the classical Petty projection inequality is established for the new Orlicz projection bodies.     

矩阵迹的几个不等式

给出了实对称矩阵的Ｈｏｌｄｅｒ不等式，Ｍｉｎｋｏｗｓｋｉ不等式和算术几何平均值不等式．     

The Discrete Planar L0-Minkowski Problem

In the discrete setting, the L₀-Minkowski problem extends the question posed and answered by the classical Minkowski's existence theorem for polytopes. In particular, the planar extension, which we address in this paper, concerns the existence of a convex polygonal body which contains the origin, whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area.     

Universal inequalities in Ehrhart theory

In this paper, we show the existence of universal inequalities for the h*-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the h*-polynomial that are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients h* ₁ and h* ₂ of the h*-vector (h* ₀, h* ₁,..., h* _d) of a lattice polytope of any degree satisfy Scott’s inequality if h* ₃ = 0.     

On pedigree polytopes and Hamiltonian cycles

In this paper we define a combinatorial object called a pedigree, and study the corresponding polytope, called the pedigree polytope. Pedigrees are in one-to-one correspondence with the Hamiltonian cycles on K_n. Interestingly, the pedigree polytope seems to differ from the standard tour polytope, Q_n with respect to the complexity of testing whether two given vertices of the polytope are nonadjacent. A polynomial time algorithm is given for nonadjacency testing in the pedigree polytope, whereas the corresponding problem is known to be NP-complete for Q_n. We also discuss some properties of the pedigree polytope and illustrate with examples.     

Stability for differential mixed variational inequalities

In this paper, an existence theorem of Carathéodory weak solution for a differential mixed variational inequality is presented under suitable conditions. Furthermore, some upper semicontinuity and continuity results concerned with the Carathéodory weak solution set mapping for the differential mixed variational inequality are given when both the mapping and the constraint set are perturbed by two different parameters.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结果;我们研究了当算子为余正仿射算子时,给出混合仿射向量变分不等式不存在例外簇的充分条件,得到混合仿射向量变分不等式弱有效解的存在性,给出了混合仿射向量变分不等式的弱有效解集为非空紧致集的充分条件.将Iusem等人(2019)在有限维空间中标量混合变分不等式解的存在性结果推广到自反Banach空间中混合向量变分不等式.     

Dynamique de la diffusion de l'erreur sur plusieurs polytopes

We discuss algorithms for scheduling, greedy for the Euclidean norm, with inputs in a family of polytopes lying in an affine space and the corresponding outputs chosen among the vertices of the respective polytopes. Such scheduling problems arise in various settings. We provide simple examples where the error remains bounded, including cases when there are infinitely many polytopes. In the case of a single polytope the boundedness of the cumulative error is known to be equivalent to the existence of an invariant region for a dynamical system in the affine space that contains this polytope. We show here that, on the contrary, no bounded invariant region can be found in affine space in general, as soon as there are at least two different polytopes. To cite this article: C. Tresser, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Minimax principles for elliptic mixed hemivariational–variational inequalities

In this paper, minimax principles are explored for elliptic mixed hemivariational–variational inequalities. Under certain conditions, a saddle-point formulation is shown to be equivalent to a mixed hemivariational–variational inequality. While the minimax principle is of independent interest, it is employed in this paper to provide an elementary proof of the solution existence of the mixed hemivariational–variational inequality. Theoretical results are illustrated in the applications of two contact problems.     

Existence theorems for generalized set-valued mixed (quasi-)variational inequalities in Banach spaces

In this paper, utilizing the properties of the generalized f -projection operator and the well-known KKM and Kakutani–Fan–Glicksberg theorems, under quite mide assumptions, we derive some new existence theorems for the generalized set-valued mixed variational inequality and the generalized set-valued mixed quasi-variational inequality in reflexive and smooth Banach spaces, respectively. The results presented in this paper can be viewed as the supplement, improvement and extension of recent results in Wu and Huang (Nonlinear Anal 71:2481–2490, 2009).     

Facets of the knapsack polytope

A necessary and sufficient condition is given for an inequality with coefficients 0 or 1 to define a facet of the knapsack polytope, i.e., of the convex hull of 0–1 points satisfying a given linear inequality. A sufficient condition is also established for a larger class of inequalities (with coefficients not restricted to 0 and 1) to define a facet for the same polytope, and a procedure is given for generating all facets in the above two classes. The procedure can be viewed as a way of generating cutting planes for 0–1 programs.     

关于矩阵迹的几个等式的注记

本文指出[1]中关于矩阵迹的H■lder和算术-几何平均不等式可从已知结论得到，而[1]中的Minkowski不等式是错误的．     

Ona-Wright convexity and the converse of Minkowski's inequality

Summary Leta (0, 1/2] be fixed. A functionf satisfying the inequalityf(ax + (1 – a)y) + f((1 – a)x + ay) f(x) + f(y), called herea-Wright convexity, appears in connection with the converse of Minkowski's inequality. We prove that every lower semicontinuousa-Wright convex function is Jensen convex and we pose an open problem. Moreover, using the fact that 1/2-Wright convexity coincides with Jensen convexity, we prove a converse of Minkowski's inequality without any regularity conditions.     

Non-projectability of polytope skeleta

We investigate necessary conditions for the existence of projections of polytopes that preserve full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal and Ziegler (2010) and the wedge product surfaces of Rörig and Ziegler (2011) and complement their results.     

BOUNDARY ELEMENT APPROXIMATION OF THE SEMI-DISCRETE PARABOLIC VARIATIONAL INEQUALITIES OF THE SECOND KIND

The boundary element approximation of the parabolic variational inequalities of the second kind is discussed. First, the parabolic variational inequalities of the second kind can be reduced to an elliptic variational inequality by using semidiscretization and implicit method in time; then the existence and uniqueness for the solution of nonlinear non-differentiable mixed variational inequality is discussed. Its corresponding mixed boundary variational inequality and the existence and uniqueness of its solution are yielded. This provides the theoretical basis for using boundary element method to solve the mixed vuriational inequality.     

Minimal covers,minimal sets and canonical facets of the posynomial knapsack polytope

An irredundant representation of the 0–1 solutions to a posynomial inequality in terms of covering constraints induced by minimal covers is given. This representation is further strengthened using extended covering constraints induced by maximal extensions of minimal covers. Necessary, sufficient, and in a special case necessary and sufficient conditions for an extended covering constraint induced by a minimal set to be a facet of the posynomial knapsack polytope are given.     

On the facets of the secondary polytope

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope.The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.     

Adjacency on the order polytope with applications to the theory of fuzzy measures

In this paper we study the adjacency structure of the order polytope of a poset. For a given poset, we determine whether two vertices in the corresponding order polytope are adjacent. This is done through filters in the original poset. We also prove that checking adjacency between two vertices can be done in quadratic time on the number of elements of the poset. As particular cases of order polytopes, we recover the adjacency structure of the set of fuzzy measures and obtain it for the set of p-symmetric measures for a given indifference partition; moreover, we show that the set of p-symmetric measures can be seen as the order polytope of a quotient set of the poset leading to fuzzy measures. From this property, we obtain the diameter of the set of p-symmetric measures. Finally, considering the set of p-symmetric measures as the order polytope of a direct product of chains, we obtain some other properties of these measures, as bounds on the volume and the number of vertices on certain cases.