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Cormac Walsh

Publications and source records attributed to Cormac Walsh.

At least 19 recordsLinked to original sources
arXiv2025

Carath\'eodory distance-preserving maps between bounded symmetric domains

We study the rigidity of maps between bounded symmetric domains that preserve the Carath\'eodory/Kobayashi distance. We show that such maps are only possible when the rank of the co-domain is at least as great as that of the domain. When the ranks are equal, and the domain is irreducible, we prove that the map is either holomorphic or antiholomorphic. In the holomorphic case, we show that the map is in fact a triple homomorphism, under the additional assumption that the origin is mapped to the origin. We exploit the large-scale geometry of the Carath\'eodory distance and use the horocompactification and Gromov product to obtain these results without requiring any smoothness assumptions on the maps.

math.CV↗
arXiv2023

Volume growth of Funk geometry and the flags of polytopes

We consider the Holmes--Thompson volume of balls in the Funk geometry on the interior of a convex domain. We conjecture that for a fixed radius, this volume is minimized when the domain is a simplex and the ball is centered at the barycenter, or in the centrally-symmetric case, when the domain is a Hanner polytope. This interpolates between Mahler's conjecture and Kalai's flag conjecture. We verify this conjecture for unconditional domains. For polytopal Funk geometries, we study the asymptotics of the volume of balls of large radius, and compute the two highest-order terms. The highest depends only on the combinatorics, namely on the number of flags. The second highest depends also on the geometry, and thus serves as a geometric analogue of the centro-affine area for polytopes. We then show that for any polytope, the second highest coefficient is minimized by a unique choice of center point, extending the notion of Santal\'o point. Finally, we show that, in dimension two, this coefficient, with respect to the minimal center point, is uniquely maximized by affine images of the regular polygon.

math.MG↗
arXiv2022

Firm non-expansive mappings in weak metric spaces

We introduce the notion of firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.

math.MG↗
arXiv2018

Flag-approximability of convex bodies and volume growth of Hilbert geometries

We introduce the flag-approximability of a convex body to measure how easy it is to approximate by polytopes. We show that the flag-approximability is exactly half the volume entropy of the Hilbert geometry on the body, and that both quantities are maximized when the convex body is a Euclidean ball. We also compute explicitly the asymptotic volume of a convex polytope, which allows us to prove that simplices have the least asymptotic volume.

math.MG↗
arXiv2018

Horofunction Compactifications of Symmetric Spaces

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.

math.DG↗
arXiv2016

Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces

We study the horofunction boundaries of Hilbert and Thompson geometries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space.

math.MG↗
arXiv2015

The horoboundary and isometry group of Thurston's Lipschitz metric

We show that the horofunction boundary of Teichmüller 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 Teichmüller spaces of different surfaces, when endowed with this metric, are not isometric, again with some possible exceptions of low genus.

math.GT↗
arXiv2013

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.

math.MG↗
arXiv2012

The asymptotic geometry of the Teichmüller metric

We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the set of Busemann points. We show that the Busemann points are exactly the limits of the Teichmüller rays, and we give a necessary and sufficient condition for a sequence of Busemann points to converge to a Busemann point. Finally, we determine the detour metric on the boundary.

math.GT↗
arXiv2011

The action of a nilpotent group on its horofunction boundary has finite orbits

We study the action of a nilpotent group G with finite generating set S on its horofunction boundary. We show that there is one finite orbit associated to each facet of the polytope obtained by projecting S into the infinite component of the abelianisation of G. We also prove that these are the only finite orbits of Busemann points. To finish off, we examine in detail the Heisenberg group with its usual generators.

math.GR↗
arXiv2009

Isometries of polyhedral Hilbert geometries

We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n>=2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex, and find that it has the collineation group as an index-two subgroup. These results confirm, for the class of polyhedral Hilbert geometries, several conjectures posed by P. de la Harpe.

math.MG↗
arXiv2008

The horofunction boundary of the Hilbert geometry

We investigate the horofunction boundary of the Hilbert geometry defined on an arbitrary finite-dimensional bounded convex domain D. We determine its set of Busemann points, which are those points that are the limits of `almost-geodesics'. In addition, we show that any sequence of points converging to a point in the horofunction boundary also converges in the usual sense to a point in the Euclidean boundary of D. We prove that all horofunctions are Busemann points if and only if the set of extreme sets of the polar of D is closed in the Painleve-Kuratowski topology.

math.MG↗
arXiv2007

Busemann points of Artin groups of dihedral type

We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterise the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.

math.GR↗
arXiv2006

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.

math.OC↗
arXiv2006

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.

math.GT↗
arXiv2005

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.

math.MG↗
arXiv2005

Minimum representing measures in Idempotent Analysis

We show that the set of max-plus measures representing a given max-plus harmonic vector has a least element. This may be viewed as an analogue of the uniqueness of the integral representation of harmonic functions in Potential Theory. As an application, we show how the distance-like functions of a metric space can be expressed in terms of the Busemann points of the metric boundary.

math.MG↗