Abstract
AbstractWe obtain characterizations of non-negative functions on $$[0,+\infty )$$
[
0
,
+
∞
)
which preserve some classes of semimetrics. In particular, one of our main results says that for a non-decreasing function $$f:[0,+\infty )\rightarrow [0,+\infty )$$
f
:
[
0
,
+
∞
)
→
[
0
,
+
∞
)
the following statements are equivalent: (i) for any semimetric space (X, d), if d satisfies the relaxed polygonal inequality, then so does $$f\circ d$$
f
∘
d
; (ii) there exist a constant $$c\geqslant 1$$
c
⩾
1
and a subadditive function $$g:[0,+\infty ) \rightarrow [0,+\infty )$$
g
:
[
0
,
+
∞
)
→
[
0
,
+
∞
)
such that $$g^{-1}\left( \{ 0 \} \right) = \{ 0 \}$$
g
-
1
{
0
}
=
{
0
}
and $$g\leqslant f \leqslant cg$$
g
⩽
f
⩽
c
g
. We also obtain a complete characterization of functions preserving regularity of a semimetric space in the sense of Bessenyei and Páles. Finally, we give another proof of the theorem of Pongsriiam and Termwuttipong on functions transforming metrics into ultrametrics.
Funder
Lodz University of Technology
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
Reference32 articles.
1. An, T.V., Dung, N.V., Kadelburg, Z., Radenović, S.: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109, 175–198 (2015)
2. An, T.V., Dung, N.V.: Answers to Kirk–Shahzad’s questions on strong $$b$$-metric spaces. Taiwan. J. Math. 20, 1175–1184 (2016)
3. Bessenyei, M., Páles, Z.: A contraction principle in semimetric spaces. J. Nonlinear Convex Anal. 18, 515–524 (2015)
4. Blumenthal, L.M.: Remarks concerning the Euclidean four-point property. Ergeb. Math. Kolloq. Wien 7, 7–10 (1936)
5. Borsík, J., Doboš, J.: Functions the composition with a metric of which is a metric. Math. Slovaca 31, 3–12 (1981)
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