Isometries, Mazur–Ulam theorem and Aleksandrov problem for non-Archimedean normed spaces

AbstractThis paper generalizes the Aleksandrov problem: the Mazur-Ulam theoremon n-G-quasi normed spaces. It proves that a one-n-distance preserving mapping is an n-isometry if and only if it has the zero-n-G-quasi preserving property, and two kinds of n-isometries on n-G-quasi normed space are equivalent; we generalize the Benz theorem to n-normed spaces with no restrictions on the dimension of spaces.

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A Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces

Abstract and Applied Analysis ◽

10.1155/2014/624920 ◽

2014 ◽

Vol 2014 ◽

pp. 1-4 ◽

Cited By ~ 1

Author(s):

J. J. Font ◽

J. Galindo ◽

S. Macario ◽

M. Sanchis

Keyword(s):

Normed Spaces ◽

Ulam Theorem

We introducefuzzy norm-preservingmaps, which generalize the concept of fuzzy isometry. Based on the ideas from Vogt, 1973, and Väisälä, 2003, we provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context: letX,Ybe fuzzy normed spaces and letf:X→Ybe a fuzzy norm-preserving surjection satisfyingf(0)=0. Thenfis additive.

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A lower bound on the waist of unit spheres of uniformly convex normed spaces

Compositio Mathematica ◽

10.1112/s0010437x1200019x ◽

2012 ◽

Vol 148 (4) ◽

pp. 1238-1264

Author(s):

Yashar Memarian

Keyword(s):

Lower Bound ◽

Strong Version ◽

Uniformly Convex ◽

Normed Spaces ◽

Localization Technique ◽

Codimension One ◽

Uniformly Convex Banach Spaces ◽

Ulam Theorem ◽

Funct Anal ◽

Compositio Math

AbstractIn this paper we give a lower bound on the waist of the unit sphere of a uniformly convex normed space by using the localization technique in codimension greater than one and a strong version of the Borsuk–Ulam theorem. The tools used in this paper follow ideas of Gromov in [Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178–215] and we also include an independent proof of our main theorem which does not rely on Gromov’s waist of the sphere. Our waist inequality in codimension one recovers a version of the Gromov–Milman inequality in [Generalisation of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263–282].

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