Families of Tight Inequalities for Polytopes

We use the Billera-Liu algebra to show how the flag f-vectors of several special classes of polytopes fit into the closed convex hull of the flag f-vectors of all polytopes. In particular, we describe inequalities that define the faces of the closed convex hull of the flag f-vectors of all d-polytopes that are spanned by the flag f-vectors of simplicial, simple, k-simplicial, and k-simple d-polytopes. We also describe inequalities that define the face of the closed convex hull of the flag f-vectors of all d-zonotopes spanned by the flag f-vectors of cubical d-zonotopes, and give an upper bound on the dimension of the span of the flag f-vectors of k-cubical zonotopes. Finally, we strengthen some previously known inequalities for flag f-vectors of zonotopes.     

Dual Representations of Convex Sets and Gâteaux Differentiability Spaces

The duality of two kinds of representations of convex sets is studied: the tangential representation of a convex body and the representations of its polar or negative polar by means of their weak^* exposed points. The equivalence of the representations is proved and a condition for their validity is obtained for individual sets (the case of arbitrary locally convex space) and for classes of sets (the case of Gâteaux differentiability locally convex space). Properties of Gâteaux differentiability locally convex spaces are studied and some examples of such spaces are given.

     

Dual Representations of Convex Sets and Gateaux Differentiability Spaces

The duality of two kinds of representations of convex sets is studied: the tangential representation of a convex body and the representations of its polar or negative polar by means of their weak^* exposed points. The equivalence of the representations is proved and a condition for their validity is obtained for individual sets (the case of arbitrary locally convex space) and for classes of sets (the case of Gâteaux differentiability locally convex space). Properties of Gâteaux differentiability locally convex spaces are studied and some examples of such spaces are given.     

Condition numbers and error bounds in convex programming

After a brief survey on condition numbers for linear systems of equalities, we analyse error bounds for convex functions and convex sets. The canonical representation of a convex set is defined. Other representations of a convex set by a convex function are compared with the canonical representation. Then, condition numbers are introduced for convex sets and their convex representations.     

Curvatures of strongly convex Kähler-Finsler manifolds

In this article, we study curvatures on a strongly convex (weakly) Kähler-Finsler manifold. First, we prove that the holomorphic sectional curvature is just half of the flag curvature in a holomorphic plane section on a strongly convex weakly Kähler-Finsler manifold. Second, we compare curvatures associated to the Rund connection with curvatures associated to the Chern-Finsler connection or the complex Berward connection on a strongly convex Kähler-Finsler manifold. Finally, we discuss relationships between flag curvatures and holomorphic bisectional curvatures, and compare two kinds of S-curvatures on a strongly convex Kähler-Finsler manifold.     

On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots. Partially supported by a KBN grant 2 P03A 017 25     

Linear Inequalities for Rank 3 Geometric Lattices

The flag Whitney numbers (also referred to as the flag f-numbers) of a geometric lattice count the number of chains of the lattice with elements having specified ranks. We give a collection of inequalities which imply all the linear inequalities satisfied by the flag Whitney numbers of rank 3 geometric lattices. We further describe the smallest closed convex set containing the flag Whitney numbers of rank 3 geometric lattices as well as the smallest closed convex set containing the flag Whitney numbers of those lattices corresponding to oriented matroids.     

Affinely Prime Dynamical Systems(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

This paper deals with representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently"minimal" actions. When the group in question is P SL(2, R), the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces. If it holds for the "universal strongly proximal space"of the group(to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

Approximation of convex bodies from chord functions

In this paper we describe a method for computing derivatives of the polar representations of the boundary of a convex body, from generalized chord functions at two points.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.     

The representations of two types of functionals on L~∞(Ω,F) and L~∞(Ω,F,P)

This article gives the representations of two types of real functionals on Z∞(Ω,F) or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.     

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups \(\mathrm {PO}(p,q)\) by considering their action on the associated pseudo-Riemannian hyperbolic space \(\mathbb {H}^{p,q-1}\) in place of the Riemannian symmetric space. Following work of Barbot and Mérigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.     

The Mayer-Vietoris andIC equations for convex polytopes

In this paper we use geometric dissection to obtain linear equations on the flag vectors on convex polytopes. These results provide new proofs and expressions of the complete system of such equations originally discovered by Bayer and Billera. The Mayer-Vietoris equation applies to a situation where two convex polytopes overlap to produce union and intersection, both convex polytopes. The operatorsI andC applied to a polytope produce the cylinder (or prism) and cone (or pyramid), respectively, with the given polytopes as base. TheIC equation relates the flag vectors of the polytopes obtained in this way. As a consequence, it becomes easier to define linear functios of the flag vector, via initial data and their law of transformation under the operatorsI andC.     

Cycles of Bott–Samelson Type for Taut Representations

Bott and Samelson constructed cycles which are concrete representativesof a basis for the Z₂-homology of the orbits ofvariationally complete representations of compact Lie groups (theseinclude isotropy representations of symmetric spaces; in this case theorbits are the so-called generalized real flag manifolds). Then theyused these cycles to show that the orbits of those representations aretaut submanifolds. We adapt the construction of Bott and Samelson to theorbits of three representations which are not variationally complete. Inthis case, it also follows that the orbits are taut.     

A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications

Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R³.     

Centered solutions for uncertain linear equations

Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we derive convex representations for united and tolerable solution sets. Secondly, to obtain centered solutions for uncertain linear equations, we develop a new method based on adjustable robust optimization (ARO) techniques to compute the maximum size inscribed convex body (MCB) of the set of the solutions. In general, the obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We use recent results from ARO to characterize for which convex bodies the obtained MCB is optimal. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input–output model, Colley’s Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

Invariance and efficiency of convex representations

We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the “smoothness” of the transformations mapping the central path of the representation to the central path of the represented optimization problem. Research of the first author was supported in part by a grant from the Faculty of Mathematics, University of Waterloo and by a Discovery Grant from NSERC. Research of the second author was supported in part by a Discovery Grant from NSERC and a PREA from Ontario, Canada.     

Geometric interpretation of Murphy bases and an application

In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give a geometric interpretation of a cellular basis of such Hecke algebras which was introduced by Murphy in the case of finite fields. We apply these results to decompose representations which arise from the space of submodules of a free module over principal ideal local rings of length two with a finite residue field.