Under the Axiom of Choice, proper polyhedral spaces admit minimizing geodesics

(Realizing a chain by a path.) Every chain $x=x_0,\dots ,x_m=y$ of length $\mathrm{\ell }$ is the vertex sequence of a path $\gamma \text{\hspace{0.17em}⁣}:[0,1]\to X$ traversing straight segments of the cells at constant speed, with $\gamma (0)=x$, $\gamma (1)=y$, $d(\gamma (s),\gamma (t))\le (t-s)\mathrm{\ell }$ for all $s\le t$, and $L(\gamma )\le \mathrm{\ell }$. Deleting one of each two consecutive equal points leaves a chain of the same length, so assume $x_{i-1}\mathrm{\ne }{x}_i$ and put $L_i:=d_{p_i}(x_{i-1},x_i)>0$ and $\mathrm{\ell }=L_1+\cdots +L_m>0$; if $\mathrm{\ell }=0$ the reduced chain is the single point $x=y$ and we take the constant path, for which all claims are immediate. For $t\in [0,1]$ define $\gamma (t)$ to be the point at fraction $(t-T_{i-1})\mathrm{/}(T_i-T_{i-1})$ of the straight segment from $x_{i-1}$ to $x_i$ inside $C_{p_i}$, where $T_i:=(L_1+\cdots +L_i)\mathrm{/}\mathrm{\ell }$; this is well defined because the segment lies in the convex cell $C_{p_i}$ [F1]. Given $s\le t$, the points $\gamma (s)$, the vertices $x_i$ strictly between the parameters $s$ and $t$, and $\gamma (t)$ form a chain whose steps lie in the traversed cells and whose length is the total traversed Euclidean distance $(t-s)\mathrm{\ell }$, because the path moves at constant speed inside each cell; hence $d(\gamma (s),\gamma (t))\le (t-s)\mathrm{\ell }$ by [F1]. Every polygonal sum of $\gamma$ over a partition is therefore at most $\mathrm{\ell }$, so the supremum over partitions gives $L(\gamma )\le \mathrm{\ell }$ [F3].

(Near-minimizing paths.) By [F7], for each $n\ge 1$ there is a chain from $x$ to $y$ of length ${\mathrm{\ell }}_n<R+1\mathrm{/}n$; apply [F4] to the countable family of nonempty sets of such chains, recording in each chain a common cell for each step (only finitely many choices per chain). This selects a sequence of chains and their realizing paths from step 1.1. Each selected chain yields a path ${\gamma }_n\text{\hspace{0.17em}⁣}:[0,1]\to X$ with ${\gamma }_n(0)=x$, ${\gamma }_n(1)=y$, $L({\gamma }_n)\le {\mathrm{\ell }}_n<R+1\mathrm{/}n$ and $d({\gamma }_n(s),{\gamma }_n(t))\le {\mathrm{\ell }}_n\mathrm{\mid }t-s\mathrm{\mid }\le (R+1)\mathrm{\mid }t-s\mathrm{\mid }$, the last inequality because $1\mathrm{/}n\le 1$ and $R\ge 0$.

(Equicontinuity and pointwise boundedness.) By step 2.1 each ${\gamma }_n$ is $(R+1)$-Lipschitz, hence continuous, and the family $({\gamma }_n)$ is equicontinuous: for a real $\epsilon >0$ the number $\delta :=\epsilon \mathrm{/}(R+1)>0$ satisfies $d({\gamma }_n(s),{\gamma }_n(t))\le (R+1)\mathrm{\mid }s-t\mathrm{\mid }<\epsilon$ for all $n$ and all $s,t$ with $\mathrm{\mid }s-t\mathrm{\mid }<\delta$. It is pointwise bounded: for every $t\in [0,1]$, $d({\gamma }_n(t),x)=d({\gamma }_n(t),{\gamma }_n(0))\le (R+1)t\le R+1$, so $\{{\gamma }_n(t):n\ge 1\}\subseteq \bar{B}(x,R+1)\subseteq B(x,R+2)$ is bounded.

(The Ascoli subsequence.) The interval $[0,1]$ is a nonempty compact metric space [F9], the target $X$ is a proper metric space [F2], and $({\gamma }_n)$ is an equicontinuous, pointwise bounded sequence of continuous maps by step 3.1, so the Ascoli-Arzela theorem [F5] — with its Choice hypothesis [F4] — gives a subsequence $({\gamma }_{n_k})$ converging uniformly to a continuous $\gamma \text{\hspace{0.17em}⁣}:[0,1]\to X$. Uniform convergence implies $\gamma (0)=x$ and $\gamma (1)=y$, since ${\gamma }_{n_k}(0)=x$ and ${\gamma }_{n_k}(1)=y$ for every $k$.

(The limit has length $R$.) By lower semicontinuity [F3], $L(\gamma )\le {\mathrm{lim\hspace{0.17em}inf}}_{k}L({\gamma }_{n_k})$. For every real $\epsilon >0$, the strictly increasing positive indices $n_k$ tend to infinity (inductively $n_k\ge k+1$), so $L({\gamma }_{n_k})\le R+1\mathrm{/}{n}_k<R+\epsilon$ for all sufficiently large $k$ [step 2.1, F10]. Every tail, including those starting earlier, contains such a term; its infimum is therefore at most $R+\epsilon$. Taking the supremum of all tail infima gives ${\mathrm{lim\hspace{0.17em}inf}}_{k}L({\gamma }_{n_k})\le R+\epsilon$ by [F10]. As this holds for every $\epsilon >0$, that limit inferior is at most $R$, and hence $L(\gamma )\le R$. Conversely the two-point partition gives $d(\gamma (0),\gamma (1))\le L(\gamma )$ [F3], that is $R\le L(\gamma )$ by step 4.1. Therefore $L(\gamma )=R<\mathrm{\infty }$.

(The minimizing geodesic.) Since $\gamma$ is continuous and rectifiable with $L(\gamma )=R$, clause (ii) of [F3] provides a continuous nondecreasing surjection $s\text{\hspace{0.17em}⁣}:[0,1]\to [0,R]$ with $s(0)=0$, $s(1)=R$, a unique $\bar{\gamma }\text{\hspace{0.17em}⁣}:[0,R]\to X$ with $\gamma =\bar{\gamma }\circ s$, and $L(\bar{\gamma }{\mathrm{\mid }}_{[r,q]})=q-r$ for all $0\le r\le q\le R$; in particular $\bar{\gamma }(0)=\gamma (0)=x$ and $\bar{\gamma }(R)=\gamma (1)=y$. Let $0\le r\le q\le R$. The chord bound gives $d(\bar{\gamma }(r),\bar{\gamma }(q))\le L(\bar{\gamma }{\mathrm{\mid }}_{[r,q]})=q-r$, and the triangle inequality for $d$ [F2] together with the chord bound on $[0,r]$ and $[q,R]$ gives $R=d(x,y)\le d(x,\bar{\gamma }(r))+d(\bar{\gamma }(r),\bar{\gamma }(q))+d(\bar{\gamma }(q),y)\le r+d(\bar{\gamma }(r),\bar{\gamma }(q))+(R-q),$ so $d(\bar{\gamma }(r),\bar{\gamma }(q))\ge q-r$. Hence $d(\bar{\gamma }(r),\bar{\gamma }(q))=q-r$ for all $r\le q$, and $\bar{\gamma }$ is a geodesic segment from $x$ to $y$ in the sense of [F6]; as $x,y$ were arbitrary, $(X,d)$ is geodesic.

(The subsequence clause.) Let now $(C^{(n)})$ be any sequence of chains from $x$ to $y$ with lengths ${\mathrm{\ell }}_n\to R$, and choose realizing paths ${\gamma }_n$ from step 1.1 using [F4] if common cells are not already specified; then $L({\gamma }_n)\le {\mathrm{\ell }}_n$ and $d({\gamma }_n(s),{\gamma }_n(t))\le {\mathrm{\ell }}_n\mathrm{\mid }t-s\mathrm{\mid }$, and ${\mathrm{\ell }}_n\le R+1$ for all sufficiently large $n$, so each ${\gamma }_n$ is $M$-Lipschitz with the common constant $M:=1+{\sup }_n{\mathrm{\ell }}_n<\mathrm{\infty }$ and the family is equicontinuous and pointwise bounded (all values lie in the bounded set $\bar{B}(x,M)$, since $d({\gamma }_n(t),x)\le {\mathrm{\ell }}_{n}t\le M$). Ascoli's theorem [F5] gives a uniformly convergent subsequence, and its limit has the same endpoints. The proof of step 5.1 applies because $L({\gamma }_{n_k})\le {\mathrm{\ell }}_{n_k}\to R$: every tail infimum is at most $R+\epsilon$ for every $\epsilon >0$, so lower semicontinuity and the chord bound give limit length exactly $R$ and whose arc-length reparametrization is, by step 6.1, a minimizing geodesic from $x$ to $y$.