Unit-sphere preserving mappings

Jung, Soon-Mo

doi:10.3336/gm.39.2.13

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Let Us Consider the Sequence2

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Cited by 2 publications

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“…In accordance with Theorem 1.1 in [12] (see also [13]), f is a linear isometry up to translations. Since f (0) = 0, then we have that…”

Section: Let Us Consider the Sequencesupporting

confidence: 71%

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The simplex dispersion ordering and its application to the evaluation of human corneal endothelia

Ayala

López-Díaz

2009

Journal of Multivariate Analysis

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a b s t r a c tA multivariate dispersion ordering based on random simplices is proposed in this paper. Given a R d -valued random vector, we consider two random simplices determined by the convex hulls of two independent random samples of sizes d + 1 of the vector. By means of the stochastic comparison of the Hausdorff distances between such simplices, a multivariate dispersion ordering is introduced. Main properties of the new ordering are studied. Relationships with other dispersion orderings are considered, placing emphasis on the univariate version. Some statistical tests for the new order are proposed. An application of such ordering to the clinical evaluation of human corneal endothelia is provided. Different analyses are included using an image database of human corneal endothelia.

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“…In accordance with Theorem 1.1 in [12] (see also [13]), f is a linear isometry up to translations. Since f (0) = 0, then we have that…”

Section: Let Us Consider the Sequencesupporting

confidence: 71%

“…Observe that the condition d ≥ 2 is necessary to apply Theorem 1.1 in [12]. If d = 1, it is possible to see that if h : R → R preserves relations between distances, then for all x ∈ R we have that h(x) = ax + b where a, b ∈ R. A sketch of the proof is the following.…”

Section: Let Us Consider the Sequencementioning

confidence: 97%