The horoboundary and isometry group of Thurston's Lipschitz metric

We show that the horofunction boundary of Teichm\"uller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichm\"uller spaces of different surfaces, when endowed with this metric, are not isometric, again with some possible exceptions of low genus.Comment: 23 pages, 5 figures. There was a mistake in one of the lemmas. Fixing it required replacing Lemmas 7.5, 7.6, and 7.7. The new version is very close to the published versio

Gauge-reversing maps on cones, and Hilbert and Thompson isometries

We show that a cone admits a gauge-reversing map if and only if it is a symmetric cone. We use this to prove that every isometry of a Hilbert geometry is a collineation unless the Hilbert geometry is the projective space of a non-Lorentzian symmetric cone, in which case the collineation group is of index two in the isometry group. We also determine the isometry group of the Thompson geometry on a cone.Comment: 36 pages, 3 figure

The horofunction boundary of finite-dimensional normed spaces

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is closed in the Painleve-Kuratowski topology.Comment: 11 pages v2. Some proofs streamlined and another example adde

How to find horizon-independent optimal strategies leading off to infinity: a max-plus approach

A general problem in optimal control consists of finding a terminal reward that makes the value function independent of the horizon. Such a terminal reward can be interpreted as a max-plus eigenvector of the associated Lax-Oleinik semigroup. We give a representation formula for all these eigenvectors, which applies to optimal control problems in which the state space is non compact. This representation involves an abstract boundary of the state space, which extends the boundary of metric spaces defined in terms of Busemann functions (the horoboundary). Extremal generators of the eigenspace correspond to certain boundary points, which are the limit of almost-geodesics. We illustrate our results in the case of a linear quadratic problem.Comment: 13 pages, 5 figures, To appear in Proc. 45th IEEE Conference on Decision and Contro

The max-plus Martin boundary

We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of max-plus harmonic functions, which give the stationary solutions of deterministic optimal control problems with additive reward. The analogue of the Martin compactification is seen to be a generalisation of the compactification of metric spaces using (generalised) Busemann functions. We define an analogue of the minimal Martin boundary and show that it can be identified with the set of limits of ``almost-geodesics'', and also the set of (normalised) harmonic functions that are extremal in the max-plus sense. Our main result is a max-plus analogue of the Martin representation theorem, which represents harmonic functions by measures supported on the minimal Martin boundary. We illustrate it by computing the eigenvectors of a class of translation invariant Lax-Oleinik semigroups. In this case, we relate the extremal eigenvectors to the Busemann points of a normed space.Comment: 37 pages; 8 figures v1: December 20, 2004. v2: June 7, 2005. Section 12 adde

Strategic Spatial Planning and the Provision of Schools: A Case Study of Cross-Sectoral Policy Coordination in the Dublin City-Region (NIRSA) Working Paper Series. No. 62.

This paper addresses the actual and potential role of strategic spatial planning in the context of educational infrastructure provision. Specifically, the paper focuses on the planning and provision of primary schools in the Dublin city-region in the context of rapid demographic and social change. The recent economic boom period has been accompanied by a rapid pace of population growth and significant shifts in the demographic composition of society in Ireland and the Dublin city-region, in particular. The analytical focus on planning for the provision of schools constitutes a critical case study of strategic spatial planning in practice. In particular, planning for school provision represents a policy domain where coordination between spatial planning and sectoral policy (i.e. education) functions is required in order to ensure the planning and provision of infrastructure to service the needs of expanding urban and peri-urban residential communities. Although, in most cases, schools are not required for development to proceed1, the need for additional school places may be particularly acute where residential development is accompanied by in-migration of households with a younger than average age profile and high proportion of young children. This paper outlines the challenges and problems associated with the practice of planning primary school provision in the Dublin city-region as well as critically assessing specific policy measures that have been introduced with the objective of improving the capacity of the state to respond to the need for new schools in areas of urban and peri-urban expansion. The analysis in this paper draws on qualitative interviews conducted by the author in 2008 and 2009