Hilbert Geometry Research Articles

Approximability of convex bodies and volume entropy in Hilbert geometry

The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary we solve the entropy upper bound conjecture in dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).

Isometries of two dimensional Hilbert geometries

We prove that every isometry between two dimensional Hilbert geometries is a projective transformation unless the domains are interiors of triangles.

Foundations of Signal Processing [Book Reviews

The landscape of signal processing books is quite populated, and it is a tall order to produce a new book that can constructively add to this landscape. This book not only accomplishes this but also stands out for its distinct, consistent, and strong personality. The book’s identity starts from a presentation that is faithful to its title since the topics covered in this book are thorough in explaining the foundations of signal processing. However, foundations can be built in many ways and with different materials. The distinct element in this book is that these foundations are built based on the use of Hilbert space geometry, which allows extending Euclidean geometric insights to signals. As such, the geometry of Hilbert spaces forms the common thread across the multiple topics explained in the book. This allows for a presentation where the topics, such as Fourier representations, sampling, interpolation, approximation, compression, and filter design, can be seamlessly unified across finite dimensions, discrete time, and continuous time.

On the Hilbert geometry of products

We prove that the Hilbert geometry of a product of convex sets is bi-lipschitz equivalent the direct product of their respective Hilbert geometries. We also prove that the volume entropy is additive with respect to product and that amenability of a product is equivalent to the amenability of each terms.

Hyperbolic is the only Hilbert geometry having circumcenter or orthocenter generally

A Hilbert geometry is hyperbolic if and only if the perpendicular bisectors or the altitudes of any trigon form a pencil. We also prove some interesting characterizations of the ellipse.

LIPSCHITZ CHARACTERISATION OF POLYTOPAL HILBERT GEOMETRIES

We study Hilbert geometries admitting similar singularities on their boundary to those of a simplex. We show that in an adapted neighborhood of those singularities, two such geometries are bi-lipschitz. As a corollary we prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.

On the Hilbert geometry of simplicial Tits sets

The moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli space, the entropy of the Hilbert metric tends to 0.

Ceva’s and Menelaus’ theorems characterize the hyperbolic geometry among Hilbert geometries

If a Hilbert geometry satisfies a rather weak version of either Ceva’s or Menelaus’ theorem for every triangle, then it is hyperbolic.

Ideal Elements in Hilbert's Geometry

Hilbert first mentioned ideal elements in his 1898-99 lectures on geometry. He described them as important, fruitful, and of frequent occurrence in mathematics, pointing to the examples of negative, irrational, imaginary, ideal and transfinite numbers. In geometry, he had in mind the examples of points, lines, and planes at infinity, whose introduction gives geometry a certain completeness, by making theorems such as those of Pappus and Desargues universally valid. In this article I will discuss how Hilbert transformed our view of the Pappus and Desargues theorems by showing that they express the underlying algebraic structure of projective geometry. I will compare this result with another of Hilbert's great contributions, his calculus of ends. By studying the ideal elements of the hyperbolic plane, Hilbert similarly extracted algebraic structure from the axioms of hyperbolic geometry. Hilbert's treatments of projective and hyperbolic geometry have another important common element: construction of real numbers. To achieve this, Hilbert has to add an axiom of continuity to the geometry axioms, but he evidently wants to show that the real numbers can be put on a geometric foundation.

Laplacian and spectral gap in regular Hilbert geometries

We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.

Two properties of volume growth entropy in Hilbert geometry

The aim of this paper is to provide two examples in Hilbert geometry which show that volume growth entropy is not always a limit on the one hand, and that it may vanish for a non-polygonal domain in the plane on the other hand.

Finitude géométrique en géométrie de Hilbert

We study the notion of geometrical finiteness for those Hilbert geometries defined by strictly convex sets with $\mathcal{C}^1$ boundary. In Gromov-hyperbolic spaces, geometrical finiteness is defined by a property of the group action on the boundary of the space. We show by constructing an explicit counter-example that this definition has to be strenghtened in order to get equivalent characterizations in terms of the geometry of the quotient orbifold, similar to those obtained by Brian Bowditch in hyperbolic geometry.

The Funk and Hilbert geometries for spaces of constant curvature

The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space \(\mathbb H ^n\) and of the sphere \(S^n\). We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of \(\mathbb R ^n\) use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces \(S^n\) and \(\mathbb H ^n\) is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert’s Problem IV in the non-Euclidean settings. Projection maps between the spaces \(\mathbb R ^n, \mathbb H ^n\) and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.

In-Phase Versus Out-of-Phase Distributed Antenna Systems for Wireless Handheld Devices

Hand loading usually degrades the performance of handset antennas. With the aim of providing a solution robust to the human presence, this paper discloses a distributed antenna system fed in-phase capable of mitigating undesirable hand loading effects while providing enhanced performance. The solution consists of a passive array of two resonant coplanar monopole antennas inspired in the Hilbert geometry. Input impedances of both monopoles are combined into a single input-output port through two transmission lines featuring equal phases. This combination allows not only improving the performance of the radiating system with respect to that attained by a single monopole antenna, but also providing a symmetric solution immune to the finger location. The solution is compared with a single monopole and a distributed antenna system fed out-of-phase. Simulated and measured results demonstrate that the proposed design is capable of providing operation in the communication standards GSM850/GSM900 as well as in the UHF part of the LTE-Advanced while maintaining the same electromagnetic performance with certain independence of the finger location.

Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ such that, for any properly open convex set $\O$ and any point $x \in \O$, any discrete group generated by a finite number of automorphisms of $\O$, which displace $x$ at a distance less than $\varepsilon_n$, is virtually nilpotent.

Asymptotic volume on Hilbert geometries

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.

Lyapunov exponents in Hilbert geometry

AbstractWe study the Lyapunov exponents of the geodesic flow of a Hilbert geometry. We prove that all of the information is contained in the shape of the boundary at the endpoint of the chosen orbit. We have to introduce a regularity property of convex functions to make this link precise. As a consequence, Lyapunov manifolds tangent to the Lyapunov splitting appear very easily. All of this work can be seen as a consequence of convexity and the flatness of Hilbert geometries.

Curvatures of spheres in Hilbert geometry

We prove that the normal curvatures of hyperspheres, the Rund curvature, and the Finsler curvature of circles in Hilbert geometry tend to 1 as radii tend to infinity.

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 for all n ≥ 2, and find that it has the collineation group as an index-two subgroup. The results confirm several conjectures of P. de la Harpe for the class of polyhedral Hilbert geometries.

Hilbert geometry for convex polygonal domains

We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to the Euclidean plane.