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
For what parameters $p\in(1,\infty)$ does (QI) hold true ?
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
Are there other normed spaces satisfying (QI), besides Eulcidian/hermitian ones ?
