Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Between art and gameness: Critical theory and computer game aesthetics

ABSTRACT This article argues that the computer game can be a locus of aesthetic form in contemporary culture. The context for understanding this claim is the decline of the artwork as bearer of form in the late 20th century, as this was understood by Adorno. Form is the enigmatic other of instrumental reason that emerges spontaneously in creative works and, in the modern era, is defined as that which makes them captivating and enigmatic yet resistant to analytic understanding. Clarification of the ways in which form is at work in game play is sought from aesthetic theory (Kant), ludology (or theory of games), and the idea of a neo-baroque entertainment culture (Ndalianis). Kant emphasized the role of play in the constitution of imaginary realms associated with aesthetic pleasure. Ludology takes play as an anthropological given differentiated histori-cally by the development of game structures. Neo-baroque theory postulates a labyrinthine, complex and de-centred entertainment culture, largely shaped by computing as a cultural practice. The article synthesizes insights from these perspectives and, drawing on ideas from Adorno and Benjamin, argues that computer games can occupy an oppositional or critical role within contempor-ary aesthetics and culture. Reflection on the constitutive processes of computer game play discloses a new place for instrumental reason within aesthetic experi-ence, as the dialectic of form and analysis migrates from traditional art materials to digital electronics