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. – Fedor Petrov Mar 27 '15 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. – Fedor Petrov Mar 27 '15 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)