Let $(M, d)$ be a metric space where
- (0) for relevance: set $M$ contains at least three distinct elements and
distance function $d : M \times M \to \mathbb R$ explicitly satisfies
(1) non-negativity: $d[ \, x, y \, ] \ge 0$ (where $x, y \in M$ are not necessarily distinct),
(2) identity of indiscernibles: $d[ \, x, y \, ] = 0 \, \iff \, x \equiv y$,
(3) symmetry: $d[ \, y, x \, ] = d[ \, x, y \, ]$, and
(4) the general triangle inequality (a.k.a. inclusive subadditivity): $d[ \, x, z \, ] \le d[ \, x, y \, ] + d[ \, y, z \, ]$ for any three (not necessarily all distinct) $x, y, z \in M$.
Now consider the following two additional conditions:
(a) the strict triangle inequality (a.k.a. exclusive subadditivity): $d[ \, x, z \, ] \lt d[ \, x, y \, ] + [ \, y, z \, ] \iff (y \not\equiv x \hbox{ and } y \not\equiv z)$,
which includes the case that $x, y, z$ are all pairwise distinct;(b) metric space $(M, d)$ is a length space;
which shall be defined here explicitly and suitably generally as follows:
$ \, $
$\forall \, x, z \in M \, | \, x \not\equiv z : (\exists \, y \in M \, | \, d[ \, x, y \, ] \lt d[ \, x, z \, ] \hbox{ and } d[ \, y, z \, ] \lt d[ \, x, z \, ]) \implies $
$ \, $
$d[ \, x, z \, ] = \text{inf}_{\{\forall \, C \subset M | \, \exists \, q \, \in \, C \, : \, q \not\equiv x \text{ and } q \not\equiv z\}}{\Large[} \, \text{sup}_{\{ n \in \mathbb N \}}{\large[} \, $
${\left\{ \text{inf}_{ \{ \forall \text{ permutations } p \, : n \rightarrow C \} } \!\! \left[ \sum_{k = 1}^n d[ \, p_{(k - 1)}, p_{(k)} \, ] \right] | {x \equiv p_{(0)} \hbox{ and } z \equiv p_{(n)} \hbox{ and } (\forall \, k \le n : p_{(k)} \in C)} \right\}} $
$ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\large]} \, {\Large]},$
where the expression part
$ \text{sup}_{\{ n \in \mathbb N \}} \left[ \, {\left\{ \text{inf}_{ \{ \forall \text{ permutations } p \, : n \rightarrow C \} } \!\! \left[ \sum_{k = 1}^n d[ \, p_{(k - 1)}, p_{(k)} \, ] \right] | {x \equiv p_{(0)} \hbox{ and } z \equiv p_{(n)} \hbox{ and } (\forall \, k \le n : p_{(k)} \in C)} \right\}} \right] $
is to be regarded as (general, possibly polygonal) length of chain $C$ between $x$ and $z$.
Surely both these conditions (a) as well as (b) are separately consistent with conditions (0) through (4) stated above, describing a corresponding metric space.
My question:
Are these conditions (a) and (b) consistent with each other?
Or in other words: Can the strict triangle inequality hold in a length space?