On mappings preserving equilateral triangles in normed spaces

被引:3
|
作者
Sikorska, Justyna [1 ]
Szostok, Tomasz [1 ]
机构
[1] Silesian Univ, Inst Math, Bankowa 14, PL-40007 Katowice, Poland
关键词
Isometry; similarity transformation; Jensen functional equation;
D O I
10.1007/s00022-006-0047-z
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let (X, vertical bar vertical bar center dot vertical bar vertical bar ) and (Y, vertical bar vertical bar center dot vertical bar vertical bar) be normed linear spaces, dim X, dim Y >= 2. We say that f : X -> Y preserves equilateral triangles if for all triples of points x, y, z epsilon X with vertical bar vertical bar x - y vertical bar vertical bar = vertical bar vertical bar y - z vertical bar vertical bar =vertical bar vertical bar x - z vertical bar vertical bar we have vertical bar vertical bar f (x) - f (y)vertical bar vertical bar = vertical bar vertical bar f (y) - f (z)vertical bar vertical bar = vertical bar vertical bar f (x) - f (z)vertical bar vertical bar. We prove that if X and Y are at least three- dimensional and f : X -> Y is surjective and preserves equilateral triangles, then f is a similarity transformation. We obtain also some new results in case where X = Y is an inner product space with dim X = 2.
引用
收藏
页码:149 / 156
页数:8
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