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 ?

• It is called Ptolemy metric space and is intensively studying. Mar 27, 2015 at 11:45
• "It is known [196] that a normed space is an inner product space if and only if it is ptolemaic." This is just found by Google, but probably it answers your question. Mar 27, 2015 at 11:50
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)