Anchored Hyperspaces and Multigraphs

Consider a multigraph $X$ as a metric space and p \in X. The anchored hyperspace at $p$ is the set  $C_p(X) =$ {A \subseteq X : p \in A, A connected and compact}. In this paper we will prove that $C_p(X)$ is a polytope if in this set is considered the Hausdorff's metric $H$. Further we will show that, if $X$ is a locally connected compact metric space such that $C_p(X)$ is a polytope for each p \in X, then $X$ must be a multigraph

Moment curves and cyclic symmetry for positive Grassmannians

We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly $\binom{n}{k}$ fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q, and show that the fixed points of a q-deformation of the cyclic shift map are precisely the critical points of the mirror-symmetric superpotential $\mathcal{F}_q$ on Gr(k,n). This follows from results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many other diverse contexts which feature moment curves and the cyclic shift map.Comment: 18 pages. v2: Minor change

Graphs of Polytopes

The graph of a polytope is the graph whose vertex set is the set of vertices of the polytope, and whose edge set is the set of edges of the polytope. Several problems concerning graphs of polytopes are discussed. The primary result is a set of bounds (Theorem 39) on the maximal size of an anticlique (sometimes called a coclique, stable set, or independent set) of the graph of a polytope based on its dimension and number of vertices. Two results concerning properties preserved by certain operations on polytopes are presented. The first is that the Gale diagram of a join of polytopes is the direct sum of the Gale diagrams of the polytopes and dually, that the Gale diagram of a direct sum of polytopes is the join of their Gale diagrams (Theorem 23). The second is that if two polytopes satisfy a weakened form of Gale's evenness condition, then so does their product (Theorem 32). It is shown, by other means, that, with only two exceptions, the complete bipartite graphs are never graphs of polytopes (Theorem 47). The techniques developed throughout are then used to show that the complete 3-partite graph K_{1,n,m} is the graph of a polytope if and only if K_{n,m} is the graph of a polytope (Theorem 49). It is then shown that K_{2,2,3} and K_{2,2,4} are never graphs of polytopes. A conjecture is then stated as to precisely when a complete multipartite graph is the graph of a polytope. Finally, a section is devoted to results concerning the dimensions for which the graph of a crosspolytope is the graph of a polytope

Triangulations

The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend

Geometrische Interpretationen und Algorithmische Verifikation von exakten Lösungen in Compressed Sensing

In an era dominated by the topic big data, in which everyone is confronted with spying scandals, personalized advertising, and retention of data, it is not surprising that a topic as compressed sensing is of such a great interest. Further the field of compressed sensing is very interesting for problems in signal- and image processing. Similarly, the question arises how many measurements are necessarily required to capture and represent high-resolution signal or objects. In the thesis at hand, the applicability of three of the most applied optimization problems with linear restrictions in compressed sensing is studied. These are basis pursuit, analysis l1-minimization und isotropic total variation minimization. Unique solutions of basis pursuit and analysis l1-minimization are considered and, on the basis of their characterizations, methods are designed which verify whether a given vector can be reconstructed exactly by basis pursuit or analysis l1-minimization. Further, a method is developed which guarantees that a given vector is the unique solution of isotropic total variation minimization. In addition, results on experiments for all three methods are presented where the linear restrictions are given as a random matrix and as a matrix which models the measurement process in computed tomography. Furthermore, in the present thesis geometrical interpretations are presented. By considering the theory of convex polytopes, three geometrical objects are examined and placed within the context of compressed sensing. The result is a comprehensive study of the geometry of basis pursuit which contains many new insights to necessary geometrical conditions for unique solutions and an explicit number of equivalence classes of unique solutions. The number of these equivalence classes itself is strongly related to the number of unique solutions of basis pursuit for an arbitrary matrix. In addition, the question is addressed for which linear restrictions do exist the most unique solutions of basis pursuit. For this purpose, upper bounds are developed and explicit restrictions are given under which the most vectors can be reconstructed via basis pursuit.In Zeiten von Big Data, in denen man nahezu täglich mit Überwachungsskandalen, personalisierter Werbung und Vorratsdatenspeicherung konfrontiert wird, ist es kein Wunder dass ein Forschungsgebiet wie Compressed Sensing von so grossem Interesse ist. Es stellt sich die Frage, wie viele Messungen tatsächlich nötig sind, um ein Signal oder ein Objekt hochaufgelöst darstellen zu können. In der vorliegenden Arbeit wird die Anwendungsmöglichkeit von drei in Compressed Sensing verwendeten Optimierungsprobleme mit linearen Nebenbedingungen untersucht. Hierbei handelt es sich namentlich um Basis Pursuit, Analysis l1-Minimierung und Isotropic Total Variation. Es werden eindeutige Lösungen von Basis Pursuit und der Analysis l1-Minimierung betrachtet, um auf der Grundlage ihrer Charakterisierungen Methoden vorzustellen, die Verifizieren ob ein gegebener Vektor exakt durch Basis Pursuit oder der Analysis l1-Minimierung rekonstruiert werden kann. Für Isotropic Total Variation werden hinreichende Bedingungen aufgestellt, die garantieren, dass ein gegebener Vektor die eindeutige Lösung von Isotropic Total Variation ist. Darüber hinaus werden Ergebnisse zu Experimenten mit Zufallsmatrizen als linearen Nebenbedingungen sowie Ergebnisse zu Experimenten mit Matrizen vorgestellt, die den Aufnahmeprozess bei Computertomographie simulieren. Weiterhin werden in der vorliegenden Arbeit verschiedene geometrische Interpretationen von Basis Pursuit vorgestellt. Unter Verwendung der konvexen Polytop-Theorie werden drei unterschiedliche geometrische Objekte untersucht und in den Zusammenhang mit Compressed Sensing gestellt. Das Ergebnis ist eine umfangreiche Studie der Geometrie von Basis Pursuit mit vielen neuen Einblicken in notwendige geometrische Bedingungen für eindeutige Lösungen und in die explizite Anzahl von Äquivalenzklassen eindeutiger Lösungen. Darüber hinaus wird der Frage nachgegangen, unter welchen linearen Nebenbedingungen die meisten eindeutigen Lösungen existieren. Zu diesem Zweck werden obere Schranken entwickelt, sowie explizite Nebenbedingungen genannt unter denen die meisten Vektoren exakt rekonstruiert werden können