A Characterization of Hyperbolic Geometry among Hilbert Geometry
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In this article, the authors characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any Hyperbolic triangle all pass through one point in one point.Abstract:
In this paper we characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any hyperbolic triangle all pass through one point.read more
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Hilbert metrics and Minkowski norms
Thomas Foertsch,Anders Karlsson +1 more
TL;DR: In this paper, it was shown that the Hilbert geometry associated to a bounded convex domain is isometric to a normed vector space if and only if $D$ is an open $n$-simplex.
Book ChapterDOI
Clustering in Hilbert simplex geometry
Frank Nielsen,Ke Sun +1 more
TL;DR: Hilbert metric in the probability simplex satisfies the property of information monotonicity, and since a canonical Hilbert metric distance can be defined on any bounded convex subset of the Euclidean space, this work considers Hilbert's projective geometry of the elliptope of correlation matrices and study its clustering performances.
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Rigidity of complex convex divisible sets
TL;DR: In this article, it was shown that the only divisible complex convex sets with $C^1$ boundary are the projective balls, which is the only convex set in real projective space that has a word hyperbolic dividing group.
Journal ArticleDOI
Ceva’s and Menelaus’ theorems characterize the hyperbolic geometry among Hilbert geometries
József Kozma,Árpád Kurusa +1 more
TL;DR: If a Hilbert geometry satisfies a rather weak version of either Ceva's or Menelaus' theorem for every triangle, then it is hyperbolic as mentioned in this paper, and if not, it is not.
References
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Book
Metric Spaces of Non-Positive Curvature
TL;DR: In this article, the authors describe the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries.
Book ChapterDOI
Extremum Problems with Inequalities as Subsidiary Conditions
TL;DR: In this paper, an extension of Lagrange's multiplier rule to the case where the subsidiary conditions are inequalities instead of equations is considered, where only extrema of differentiable functions of a finite number of variables will be considered.
An Elementary Introduction to Modern Convex Geometry
TL;DR: In this paper, the Brunn-Minkowski inequality and its extensions are discussed and the Reverse Isoperimetric Problem is solved. But the central limit theorem and large deviation inequalities are not considered.
Book
Convex and Discrete Geometry
TL;DR: Convex Functions and Convex Bodies as discussed by the authors have been studied in the context of convex polytopes for geometry of numbers and aspects of discrete geometry.