The ptolemaic inequality in Hilbert geometries

Kay, David C.

doi:10.2140/pjm.1967.21.293

Cited by 20 publications

(16 citation statements)

References 3 publications

(8 reference statements)

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“…In 1963, in his doctoral thesis, Kay showed that a Riemannian manifold is non-positively curved if and only if it is locally Ptolemaic (for more details, see [21]) and that a Finsler space which is locally Ptolemaic is Riemannian. We show that the quadrilateral inequality condition implies the Ptolemaic inequality.…”

Section: IVmentioning

confidence: 99%

Quasilinearization and curvature of Aleksandrov spaces

2008

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We present a new distance characterization of Aleksandrov spaces of nonpositive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine-a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space (M, ρ) is an Aleksandrov 0 domain (also known as a CAT (0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in M. We also observe that a geodesically connected metric space (M, ρ) is an 0 domain if and only if, for every quadruple of points in M, the quadrilateral inequality (known as Euler's inequality in R 2 ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an 0 domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.

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Section: IVmentioning

confidence: 99%

Quasilinearization and curvature of Aleksandrov spaces

2008

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“…Une part importante des travaux récent concernant ces géométries consiste à étudier les différences et liens qu'elles peuvent partager avec la géo-métrie hyperbolique. Ainsi, si K n'est pas un ellipsoïde, la géométrie n'est pas riemannienne, voir D.C. Kay [19,Corollary 1]. Ce dernier résultat étant dû au fait que parmi les espaces vectoriels de dimension finie, bon nombre de notions de courbures sont satisfaites uniquement par les espaces euclidiens (voir aussi P. Kelly & L. Paige [22], P. Kelly & E. Strauss [20,21]).…”

Section: Introduction Et Présentation Des Résultatsunclassified

Sur l’entropie volumique des géométries de Hilbert

Vernicos

2008

Séminaire De Théorie Spectrale Et Géométrie

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“…A metric midpoint of Pl and P2 will be denoted by m [pl, p2] In [10], metric spaces which satisfy inequality (9) locally are said to have feeble non-positive median space curvature. However, unlike (8), the much weaker inequality (9) is valid locally in classical spaces of positive curvature.…”

Section: Comparison Of Metric Inequalitiesmentioning

confidence: 99%

“…For each point p in Z] the spherical neighborhood U(p; r), where r2k ~ (~r]4) 2, satisfies the inequalities(9),(10),(11) with tz = ~r]2 and ~ ~ Tf, Proof We do not attempt here to give the best values of the constants tz and ~. The hyperbolic and Euclidean cases (k ~< 0) are covered by Lemmas 4.1, 4.2 and 4.3 with r = oo.…”

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confidence: 99%

“…For each point p in Z] the spherical neighborhood U(p; r), where r2k ~ (~r]4) 2, satisfies the inequalities(9),(10),(11) with tz = ~r]2 and ~ ~ Tf, Proof We do not attempt here to give the best values of the constants tz and ~. Due to Lemma 4.3, we need only show that inequalities(9) and(10)are satisfied in U(p; zrR/4) with/z = zr]2. Therefore, we may assume that k > 0 and consequently Z~ is a spherical space with radius R = k-1/2.…”

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confidence: 99%

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