In a Euclidian space (Hermitian as well), say $\ell^2\_n$, the following inequality holds true
$$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2\_n.$$
In other words, the function
$$\delta:=\frac{|b-a|}{|a|\cdot|b|}$$
is a distance over $\ell^2\_n\setminus\{0\}$.


The proof consists in applying the triangle inequality to the vectors $Ia:=|a|^{-2}a$, $Ib$, $Ic$, obtained by applying the inversion with respect to the unit sphere:
$$\delta(a,b)=|Ib-Ia|.$$


It turns out that (QI) is false in $\ell^1\_n$ when $n\ge2$. A counter-example is given by the choice 
$$a=\begin{pmatrix} 1 \\ 0 \end{pmatrix},\quad b=\begin{pmatrix} 1 \\ 1 \end{pmatrix},\quad c=\begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$
This is amazing, because (QI) can be used to prove Hlawka's inequality in $\ell^2\_n$, an inequality that turns out to be true also in $\ell^1\_n$ (no contradiction, of course).


Because $\ell^\infty\_2$ is isometric to $\ell^1\_2$, (QI) is false in $\ell^\infty\_n$ as well for $n\ge2$. Rotating the above triplet by $-\frac\pi4$, we get the following counter-example
$$a'=\begin{pmatrix} 1 \\ 1 \end{pmatrix},\quad b'=\begin{pmatrix} 2 \\ 0 \end{pmatrix},\quad c'=\begin{pmatrix} 1 \\ -1 \end{pmatrix}.$$
A natural question is



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> For what parameters $p\in(1,\infty)$ does (QI) hold true ?
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Actually, the triplet $(a,b,c)$ provides a counter-example for $p<2$, while $(a',b',c')$ is a counter-example for $p>2$. Therefore, only $\ell^2$ satifies (QI). This let me asking 



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> Are there other normed spaces satisfying (QI), besides Eulcidian/hermitian ones ?
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A "quadratic" triangular inequality
Metric space $(X,\rho)$ satisfying Ptolemy inequality $\rho(a,b)\rho(c,d)+\rho(b,c)\rho(a,d)\geq \rho(a,c)\rho(b,d)$ is called ptolemaic space. A normed ptolemaic space must be inner product space. 
Reference: I.J. Schoenberg, A remark on M. M. Day’s characterization of innerproduct spaces and a conjecture of L. M. Blumenthal. Proc. Am. Math. Soc. 3, 961–964 (1952)


