Is $z$ a point on the line $[x,y]={tx+(1-t)y: tin[0,1]}$?

Question

In a normed space $M$ with $x,y,zin M$, i  need show if this statement is true or false.
$|x-y |= |x-z | + |z-y |$ if and only if $z$ is a point on the line $[x,y]={tx+(1-t)y: tin[0,1]}$
I showed the direction $(Leftarrow)$ and think that in general the other direction $(Rightarrow)$ is not true but can't find an example for this.
I hope you can help me.
Thank you!

José Carlos Santos · Accepted Answer

In $Bbb R^2$, consider the norm $|(a,b)|_infty=max{|a|,|b|}$. Then, if $x=(-1,0)$, $y=(1,0)$, and $z=(0,1)$, you have$$|x-y|_infty=2=overbrace{|x-z|_infty}^{phantom1=1}+overbrace{|z-y|_infty}^{phantom1=1}.$$However,$$znotin{tx+(1-t)ymid tin[0,1]}.$$

Correct answer by José Carlos Santos on January 15, 2021

Is $z$ a point on the line $[x,y]={tx+(1-t)y: tin[0,1]}$?

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