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Convex Function Of A Metric

TOPO-K91BFX

A function $\varphi:[0,\infty)\to \mathbb{R}\,$ is convex if:

$$ \varphi(\lambda a + (1-\lambda) b) \leq \lambda \varphi(a) + (1-\lambda) \varphi(b), \ \mbox{ for all } a,b\in [0,\infty), \mbox{ and all } \lambda\in[0,1]$$

Let $d\,$ be a metric on $X$.

Which of the following expressions DO NOT ALWAYS define a metric on $X\,$?

A

$\max\{d,1\}$.

B

$\min\{d,1\}$.

C

$\varphi(d)$, where $\varphi$ is convex and non-negative.

D

$\varphi(d)$, where $\varphi$ is convex and $\varphi(0)=0$.

E

$\varphi(d)$, where $\varphi$ is convex, $\varphi(0)=0$, and for some $\epsilon>0$, $\varphi(s)>0$ for all $s\in (0,\epsilon)$.