On the Combinatorial Complexity of Approximating Polytopes

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error parameter $\varepsilon > 0$, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon \cdot \mathrm{diam}(K)$. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $O(1/\varepsilon^{(d-1)/2})$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $\tilde{O}(1/\varepsilon^{(d-1)/2})$, where $\tilde{O}$ conceals a polylogarithmic factor in $1/\varepsilon$. This is a significant improvement upon the best known bound, which is roughly $O(1/\varepsilon^{d-2})$. Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.Comment: In Proceedings of the 32nd International Symposium Computational Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and Computational Geometr

«The war is a racket!» The emergence of the libertarian discourse about world war I in the United States

"It is not a coincidence that the century of war coincided with the century of central banking,” wrote Ron Paul, the libertarian candidate "sensation" for the presidential elections in 2008 and 2012, in the book End the Fed. This discussion explores in short, the powerful pamphlet by Major General Smedley Butler, "War is a Racket", demonstrating, specifically, who profited economically and who, in turn, bore the weight and violence of WW1, assuming that a war is never fought with the acquiescence of the population. However, this monograph goes further, looking for a reinterpretation of the official American history of the First World War through the lens of libertarian discourse. The aim is thus to understand, from another perspective, the fundamental cause of the paradigm shift from nonintervention to intervention taking place during this war, linking it to the project which led to the creation of the League of Nations and the growing importance of the US in the world. Finally, a fundamental connection will be established, exploring the theories argued in the book A Foreign Policy of Freedom, between the policies of Woodrow Wilson and the foreign policy of the United States throughout the 20th century and the beginning of the 21st

(un)childhood: performing the voices and times of childhood through relational video-making

This practice-based PhD is comprised of two interrelated elements: (i) ‘(un)childhood’, a 53’ video-essay shown on two screens; and (ii) a 58286 word written thesis. The project, which is contextualised within the tradition of artists working with their own children on time-based art projects, explores a new approach to timebased artistic work about childhood. While Stan Brakhage (1933-2003), Ernie Gher (1943-), Erik Bullot (1963-) and Mary Kelly (1941-) all documented, photographed and filmed their children over a period of years to produce art projects (experimental films and a time-based installation), these projects were implicitly underpinned by a construction of childhood in which children, shown as they grow, represent the abstract primitive subject. The current project challenges the convention of representing children entirely from the adult’s point of view, as aesthetic objects without a voice, as well as through the artist’s chronological approach to time. Instead, this project focuses on the relational joining of the child’s and adult’s points of view. The artist worked on a video project with her own son over a four-and-a-half year period (between the ages of 5 and 10) through which she developed her ‘relational video-making’ methodology. The video-essay (un)childhood performs the relational voices of childhood as resulting from the verbal interactions of both children and adults. The non-chronological nature of(un)childhood offers an alternative to the linear-temporal approach to the representation of childhood. Through montage and a number of literal allusions to time in its dialogue, (un)childhood performs the relational times of childhood by combining children’s lives in the present with the temporal dimensions that have traditionally constructed childhood: past, future and timeless

The eigenpairs of a Sylvester-Kac type matrix associated with a simple model for one-dimensional deposition and evaporation

A straightforward model for deposition and evaporation on discrete cells of a finite array of any dimension leads to a matrix equation involving a Sylvester-Kac type matrix. The eigenvalues and eigenvectors of the general matrix are determined for an arbitrary number of cells. A variety of models to which this solution may be applied are discussed.Comment: 7 pages, no figure