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-Isometries in Euclidean spaces

✍ Scribed by Igor A. Vestfrid

Publisher: Elsevier Science
Year: 2005
Tongue: English
Weight: 128 KB
Volume: 63
Category: Article
ISSN: 0362-546X
DOI: 10.1016/j.na.2005.05.036

No coin nor oath required. For personal study only.

✦ Synopsis

Let 0 < r R and A be a subset of the n-dimensional Euclidean space E n , which is contained in B(x 0 , R) and contains points x 0 , x 0 + re 1 , . . . , x 0 + re n , where the vectors {e i } n i=1 are orthonormal. We show that if f : A → E n is an -isometry, then there is an affine isometry U such that f (x)-U(x)

(n) R/r for all x in A, where (n) = Cn. Moreover, if A is star-shaped with respect to x 0 and contains B(x 0 , r), then (n) = C log(n + 1). Both results are sharp.

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