Convex geometries are combinatorial structures; they capture in an abstract way the essential features of convexity in Euclidean space, graphs or posets for instance. A convex geometry consists of a finite ground set plus a collection of subsets, called the convex sets and satisfying certain axioms. In this work, we study two natural problems on convex geometries. First, we consider the maximum-weight convex set problem. After proving a hardness result for the problem, we study a special family of convex geometries built on split graphs. We show that the convex sets of such a convex geometry relate to poset convex geometries constructed from the split graph. We discuss a few consequences, obtaining a simple polynomial-time algorithm to solve the problem on split graphs. Next, we generalize those results and design the first polynomial-time algorithm for the maximum-weight convex set problem in chordal graphs. Second, we consider the realizability problem. We show that deciding if a given convex geometry (encoded by its copoints) results from a point set in the plane is ER-hard. We complete our text with a brief discussion of potential further work. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Thesis (MSc (Mathematics))--University of Stellenbosch, 2008. / Let T be a non-empty set, A a u-algebra of subsets of T and u : .A -+ Rn a bounded, countably additive measure. A set E E A is called an atom with respect to u if u(E)=/F 0 and, if F E A, FeE, then u(F) = u(E) or u(F) = 0; the measure u is atomic if there exists at least one atom (with respect to u) in A. If no such atom (with respect to u) exists in A, then u is called non-atomic. In 1940 the Russian mathematician A. A. Lyapunov published the Convexity Theorem. According to this theorem the range 'R.{u) of a bounded, finite-dimensional measure u is compact and, in the non-atomic case, convex. Since 1940 much has been published on different aspects of the range of a vector-measure. These aspects range from new and shorter proofs of the Convexity Theorem and the usefulness of it in diverse fields, to research about the geometrical characteristics of the range by using other familiar theorems, like Krein-Milman and Radon-Nikodym. In the survey at hand the Convexity Theorem in itself is studied. Applications in different fields will be looked at as well as pieces about the history of the people and the ideas involved in the development of the theorem.

Cette thèse est consacrée aux propriétés géométriques des opérations de tores algébriques dans les variétés affines. Elle est issue de trois prépublications qui correspondent aux points (1), (2), (3) ci-après. Soit X une variété affine munie d’une opération d’un tore algébrique T. Nous appelons complexité la codimension de l’orbite générale de T dans X. Sous l’hypothèse de normalité et lorsque le corps de base est algébriquement clos de caractéristique 0, la variété X admet une description combinatoire en termes de géométrie convexe. Cette description, obtenue en 2006 par Altmann et Hausen, généralise celle classique des variétes toriques. Notre but consiste à étudier des problèmes nouveaux concernant les propriétés algébriques et géométriques de X lorsque l’operation de T dans X est de complexité 1. (1) Dans la première partie, un résultat donne une manière explicite de déterminer la clôture intégrale de toute variété affine définie sur un corps algébriquement clos de caractérisque 0 munie d’une opération de T de complexité 1 en termes de la description combinatoire d’Altmann-Hausen. Comme application, nous donnons une classification complète des idéaux intégralement clos homogènes de l’algèbre des fonctions régulières de X et généralisons un théorème de Reid-Roberts-Vitulli sur la description de certains idéaux normaux de l’algèbre des polynômes à plusieurs variables. (2) Les calculs de la première partie suggèrent une démonstration de la validité de la présentation d’Altmann-Hausen sur un corps quelconque dans le cas de complexité 1. Ce qui est fait dans la deuxième partie. Dans la situation non déployée, la descente galoisienne d’une variété affine normale munie d’une opération d’un tore algébrique de complexité 1 est décrite par un nouvel objet combinatoire que nous appelons diviseur polyédral Galois stable. (3) Dans la troisième partie, lorsque que le corps de base est parfait, nous classifions toutes les opérations du groupe additif dans X normalisées par l’action de T de complexité 1. Cette classification généralise des travaux classiques de Flenner et Zaidenberg dans le cas des surfaces et de Liendo dans le cas où le corps ambiant est algébriquement clos de caractéristique 0. / This thesis is devoted to the study of geometric properties of affine algebraic varieties endowed with an action of an algebraic torus. It comes from three preprints which correspond to the indicated points (1), (2), (3). Let X be an affine variety equipped with an action of the algebraic torus T. The complexity of the T-action on X is the codimension of the general T-orbit. Under the assumption of normality and when the ground field is algebraically closed of characteristic 0, the variety X admits a combinatorial description in terms of convex geometry. This description obtained by Altmann and Hausen in the year 2006 generalizes the classical one for toric varieties. Our purpose is to investigate new problems on the algebraic and geometric properties of the variety X when the T-action on X is of complexity 1. (1) In the first part, a result gives an effective method to determine the integral closure of any affine variety defined over an algebraically field of characteristic 0 with a T-action of complexity 1 in terms of the combinatorial description of Altmann-Hausen. As an application, we provide an entire classification of the homogeneous integrally closed ideals of the algebra of regular functions on X and generalize the Reid-Roberts-Vitulli's theorem on the description of certain normal ideals of the polynomial algebra. (2) The calculations of the first part suggest a proof of the validity of the presentation of Altmann-Hausen in the case of complexity 1 over an arbitrary ground field. This is done in the second part of this thesis. In the non-split situation, the Galois descent of normal affine varieties with a T-action of complexity 1 is described by a new combinatorial object which we call a Galois invariant polyhedral divisor. (3) In the third part, when the base field is perfect, we classify all the actions of the additive group on X normalized by the T-action of complexity 1. This classification generalizes classical works of Flenner and Zaidenberg in the surface case and of Liendo when the base field is algebraically closed of characteristic 0.

Cette thèse est consacrée à l'étude des mesures convexes ainsi qu'aux analogies entre la théorie de Brunn-Minkowski et la théorie de l'information. Je poursuis les travaux de Costa et Cover qui ont mis en lumière des similitudes entre deux grandes théories mathématiques, la théorie de Brunn-Minkowski d'une part et la théorie de l'information d'autre part. Partant de ces similitudes, ils ont conjecturé, comme analogue de la concavité de l'entropie exponentielle, que la racine n-ième du volume parallèle de tout ensemble compact de $R^n$ est une fonction concave, et je résous cette conjecture de manière détaillée. Par ailleurs, j'étudie les mesures convexes définies par Borell et je démontre pour ces mesures une inégalité renforcée de type Brunn-Minkowski pour les ensembles convexes symétriques. Cette thèse se décompose en quatre parties. Tout d'abord, je rappelle un certain nombre de notions de base. Dans une seconde partie, j'établis la validité de la conjecture de Costa-Cover sous certaines conditions et je démontre qu'en toute généralité, cette conjecture est fausse en exhibant des contre-exemples explicites. Dans une troisième partie, j'étends les résultats positifs de cette conjecture de deux manières, d'une part en généralisant la notion de volume et d'autre part en établissant des versions fonctionnelles. Enfin, je prolonge des travaux récents de Gardner et Zvavitch en améliorant la concavité des mesures convexes sous certaines hypothèses telles que la symétrie / This thesis is devoted to the study of convex measures as well as the relationships between the Brunn-Minkowski theory and the Information theory. I pursue the works by Costa and Cover who highlighted similarities between two fundamentals inequalities in the Brunn-Minkowski theory and in the Information theory. Starting with these similarities, they conjectured, as an analogue of the concavity of entropy power, that the n-th root of the parallel volume of every compact subset of $R^n$ is concave, and I give a complete answer to this conjecture. On the other hand, I study the convex measures defined by Borell and I established for these measures a refined inequality of the Brunn-Minkowski type if restricted to convex symmetric sets. This thesis is split in four parts. First, I recall some basic facts. In a second part, I prove the validity of the conjecture of Costa-Cover under special conditions and I show that the conjecture is false in such a generality by giving explicit counterexamples. In a third part, I extend the positive results of this conjecture by extending the notion of the classical volume and by establishing functional versions. Finally, I generalize recent works of Gardner and Zvavitch by improving the concavity of convex measures under different kind of hypothesis such as symmetries

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The results presented in this dissertation refer to the geometry of Minkowski spaces, i.e., of real finite-dimensional Banach spaces. First we study geometric properties of radial projections of bisectors in Minkowski spaces, especially the relation between the geometric structure of radial projections and Birkhoff orthogonality. As an application of our results it is shown that for any Minkowski space there exists a number, which plays somehow the role that $\sqrt2$ plays in Euclidean space. This number is referred to as the critical number of any Minkowski space. Lower and upper bounds on the critical number are given, and the cases when these bounds are attained are characterized. Moreover, with the help of the properties of bisectors we show that a linear map from a normed linear space $X$ to another normed linear space $Y$ preserves isosceles orthogonality if and only if it is a scalar multiple of a linear isometry. Further on, we examine the two tangent segments from any exterior point to the unit circle, the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, the distances from the normalization of the sum of two unit vectors to those two vectors, and the extension of the notions of orthocentric systems and orthocenters in Euclidean plane into Minkowski spaces. Also we prove theorems referring to chords of Minkowski circles and balls which are either concurrent or parallel. All these discussions yield many interesting characterizations of the Euclidean spaces among all (strictly convex) Minkowski spaces. In the final chapter we investigate the relation between the length of a closed curve and the length of its midpoint curve as well as the length of its image under the so-called halving pair transformation. We show that the image curve under the halving pair transformation is convex provided the original curve is convex. Moreover, we obtain several inequalities to show the relation between the halving distance and other quantities well known in convex geometry. It is known that the lower bound for the geometric dilation of rectifiable simple closed curves in the Euclidean plane is $\pi/2$, which can be attained only by circles. We extend this result to Minkowski planes by proving that the lower bound for the geometric dilation of rectifiable simple closed curves in a Minkowski plane $X$ is analogously a quarter of the circumference of the unit circle $S_X$ of $X$, but can also be attained by curves that are not Minkowskian circles. In addition we show that the lower bound is attained only by Minkowskian circles if the respective norm is strictly convex. Also we give a sufficient condition for the geometric dilation of a closed convex curve to be larger than a quarter of the perimeter of the unit circle.

Orientadores: Carlos Eduardo Durán Fernandez, Márcio Antônio de Faria Rosa / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-16T14:48:33Z (GMT). No. of bitstreams: 1 ManoOtero_Diego_M.pdf: 1221591 bytes, checksum: 33ae6e3b671523a9602f3398e14d4fb7 (MD5) Previous issue date: 2010 / Resumo: A motivação inicial deste trabalho foi tentar relacionar os conceitos de geometria de Finsler com situações físicas que temos uma certa dependência de direções no nosso espaço. Apresentamos o conceito do cálculo variacional em variedades e sua relação com as geodésicas. Estudamos também o operador laplaciano ?? para espaços de Minkowski, que generaliza o caso Euclideano, e mais especificamente o problema...Observação: O resumo, na íntegra poderá ser visualizado no texto completo da tese digital / Abstract: The initial motivation of this study was to try to relate the concepts of Finsler geometry with physical situations where we have a certain dependence on the directions of our space. We introduce the concept of variational calculus on manifolds and their relationship with the geodesics. We also studied the Laplacian operator ?? in Minkowski space, which generalizes the Euclidean case, and more specifically the problem ...Note: The complete abstract is available with the full electronic digital thesis or dissertations. / Mestrado / Geometria / Mestre em Matemática

Etant donné trois entiers k, s et a, nous prouvons dans le premier chapitre qu'il existe un graphe k-régulier fini (resp. un graphe k-régulier connexe fini) dont le groupe d'automorphismes a exactement s orbites sur l'ensemble des sommets et a orbites sur l'ensemble des arêtes si et seulement si<p><p>(s,a)=(1,0) si k=0,<p>(s,a)=(1,1) si k=1,<p>s=a>0 si k=2,<p>0< s <= 2a <= 2ks si k>2.<p><p>(resp.<p>(s,a)=(1,0) si k=0,<p>(s,a)=(1,1) si k=1 ou 2,<p>s-1<=a<=(k-1)s+1 et s,a>0 si k>2.)<p><p>Nous étudions les polyèdres convexes de R³ dans le second chapitre. Pour tout polyèdre convexe P, nous notons Isom(P) l'ensemble des isométries de R³ laissant P invariant. Si G est un sous-groupe de Isom(P), le f_G-vecteur de P est le triple d'entiers (s,a,f) tel que G ait exactement s orbites sur l'ensemble sommets de P, a orbites sur l'ensemble des arêtes de P et f orbites sur l'ensemble des faces de P. Remarquons que (s,a,f) est le f_{id}-vecteur (appelé f-vecteur dans la littérature) d'un polyèdre si ce dernier possède exactement s sommets, a arêtes et f faces. Nous généralisons un théorème de Steinitz décrivant tous les f-vecteurs possibles. Pour tout groupe fini G d'isométries de R³, nous déterminons l'ensemble des triples (s,a,f) pour lesquels il existe un polyèdre convexe ayant (s,a,f) comme f_G-vecteur. Ces résultats nous permettent de caractériser les triples (s,a,f) pour lesquels il existe un polyèdre convexe tel que Isom(P) a s orbites sur l'ensemble des sommets, a orbites sur l'ensemble des arêtes et f orbites sur l'ensemble des faces.<p><p>La structure d'incidence I(P) associée à un polyèdre P consiste en la donnée de l'ensemble des sommets de P, l'ensemble des arêtes de P, l'ensemble des faces de P et de l'inclusion entre ces différents éléments (la notion de distance ne se trouve pas dans I(P)). Nous déterminons également l'ensemble des triples d'entiers (s,a,f) pour lesquels il existe une structure d'incidence I(P) associée à un polyèdre P dont le groupe d'automorphismes a exactement s orbites de sommets, a orbites d'arêtes et f orbites de sommets. / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished

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