## Inner l^1 metric on a simplicial complex as Pullback

Let $X=(V,S)$ be a finite simplicial complex and let $n:=|V|$. The $l^1$-metric $d$ on the realization $|X|$ is defined to be the restriction of the $l^1$ metric on $\mathbb{R}^n$ to $|X|$ using the usual embedding (i.e. send a vertex to the corresponding base element and extend this map linearly on each simplex).

Now the resulting metric might not be inner; as the example of two triangles glued along one side shows. However in this example it is possible to modify the embedding, such that the restriction of the $l^1$ metric on $\mathbb{R}^n$ is the inner metric associated to $d$.

Let $ABC,BCD$ be two triangles. Send $A,B,C$ to the standart base of $\mathbb{R}^3$. $D$ should be send to the reflection of $A$ at the point $(B+C)/2$. Then the pullback of the $l^1$-metric on $\mathbb{R}^3$ is the metric I am looking for.

So the question is: Is it always possible to modify the embedding in such a way, i.e. Is every finite simplicial complex equipped with the inner $l^1$ metric isometric to a subspace of some $\mathbb{R}^n$ (equipped with the $l^1$ metric) ?

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