Length in a metric target: lower semicontinuity and arc-length reparametrization

(Chord bound, additivity at the initial point, monotonicity.) For $a\le u\le v\le b$ if $u=v$ both the chord and the length are $0$ by the singleton convention and [F1]; if $u<v$ the sequence $u,v$ is a partition of $[u,v]$ with polygonal sum $d(\gamma (u),\gamma (v))$, so $d(\gamma (u),\gamma (v))\le L(\gamma {\mathrm{\mid }}_{[u,v]})$ because $L(\gamma {\mathrm{\mid }}_{[u,v]})$ is the least upper bound of all polygonal sums [F3]. For $a\le v\le w\le b$ let first $P$ be a partition of $[a,w]$; inserting $v$ if it is absent gives a partition $P^{\mathrm{\prime }}$ of $[a,w]$ whose polygonal sum is at least that of $P$ by the triangle inequality [F1], and which is the union of a partition of $[a,v]$ and a partition of $[v,w]$. Hence every polygonal sum of $\gamma {\mathrm{\mid }}_{[a,w]}$ is at most $L(\gamma {\mathrm{\mid }}_{[a,v]})+L(\gamma {\mathrm{\mid }}_{[v,w]})$, so $L(\gamma {\mathrm{\mid }}_{[a,w]})\le L(\gamma {\mathrm{\mid }}_{[a,v]})+L(\gamma {\mathrm{\mid }}_{[v,w]})$ [F3]. Conversely, if both summands are finite, then for every real $\epsilon >0$ there are partitions of $[a,v]$ and of $[v,w]$ with sums exceeding $L(\gamma {\mathrm{\mid }}_{[a,v]})-\epsilon \mathrm{/}2$ and $L(\gamma {\mathrm{\mid }}_{[v,w]})-\epsilon \mathrm{/}2$, and their union is a partition of $[a,w]$, so $L(\gamma {\mathrm{\mid }}_{[a,w]})\ge L(\gamma {\mathrm{\mid }}_{[a,v]})+L(\gamma {\mathrm{\mid }}_{[v,w]})-\epsilon$; as the real $\epsilon >0$ is arbitrary, [F7] gives $L(\gamma {\mathrm{\mid }}_{[a,w]})\ge L(\gamma {\mathrm{\mid }}_{[a,v]})+L(\gamma {\mathrm{\mid }}_{[v,w]})$. If $L(\gamma {\mathrm{\mid }}_{[a,v]})=+\mathrm{\infty }$, then every partition of $[a,v]$ extends by $w$ to a partition of $[a,w]$ with sum no smaller, since the added chord is nonnegative, so $L(\gamma {\mathrm{\mid }}_{[a,w]})\ge L(\gamma {\mathrm{\mid }}_{[a,v]})=+\mathrm{\infty }$ and the two sides agree; and if $L(\gamma {\mathrm{\mid }}_{[a,v]})<\mathrm{\infty }$ while $L(\gamma {\mathrm{\mid }}_{[v,w]})=+\mathrm{\infty }$, then for every real $M$ there is a partition of $[v,w]$ with sum exceeding $M$, and adjoining any fixed partition of $[a,v]$ yields a partition of $[a,w]$ with sum exceeding $M$, so $L(\gamma {\mathrm{\mid }}_{[a,w]})=+\mathrm{\infty }$. The degenerate cases $v=a$ or $v=w$ follow directly from the singleton convention. This proves additivity, and monotonicity of $s$ follows from $s(w)=s(v)+L(\gamma {\mathrm{\mid }}_{[v,w]})\ge s(v)$ for $v\le w$ because lengths are suprema of sums of nonnegative terms [F1, F3].

(Lower semicontinuity.) Suppose $u\in \bar{\mathbb{R}}$ is an upper bound of the tail infima ${\mathrm{\ell }}_N:={\inf }_{k\ge N}L({\gamma }_k)$ of [F5]; I show $u\ge L(\gamma )$. Assume $u<L(\gamma )$. Then $u$ is a real number, because $u=+\mathrm{\infty }$ contradicts $u<L(\gamma )\le +\mathrm{\infty }$ and $u=-\mathrm{\infty }$ is impossible as all $L({\gamma }_k)\ge 0$ forces every ${\mathrm{\ell }}_N\ge 0$ [F1, F3]. Since $L(\gamma )$ is the least upper bound of the polygonal sums of $\gamma$ [F3] and $u<L(\gamma )$, some partition $P$ of $[a,b]$ has polygonal sum $S(P)>u$. By [F7] choose a natural number $n\ge 1$ with $1\mathrm{/}n<S(P)-u$. For each partition point $t_i$, the convergence ${\sup }_{t}d({\gamma }_k(t),\gamma (t))\to 0$ gives $d({\gamma }_k(t_i),\gamma (t_i))\to 0$ [F4]; consequently $S_k(P):={\sum }_{i}d({\gamma }_k(t_{i-1}),{\gamma }_k(t_i))$ converges to $S(P)$ by [F2] and [F6]. So there is $K$ with $S_k(P)>S(P)-1\mathrm{/}n>u$ for all $k\ge K$. Since $S_k(P)\le L({\gamma }_k)$ by [F3], this gives ${\mathrm{\ell }}_K={\inf }_{k\ge K}L({\gamma }_k)\ge S(P)-1\mathrm{/}n>u$, contradicting that $u$ is an upper bound of the ${\mathrm{\ell }}_N$. Hence every upper bound $u$ of the tail infima satisfies $u\ge L(\gamma )$, and since the least such upper bound is ${\mathrm{lim\hspace{0.17em}inf}}_{k}L({\gamma }_k)$ [F5], $L(\gamma )\le {\mathrm{lim\hspace{0.17em}inf}}_{k}L({\gamma }_k)$.

(The arclength function.) From now on assume that $\gamma$ is continuous and that $L:=L(\gamma )<\mathrm{\infty }$; write $s(t):=L(\gamma {\mathrm{\mid }}_{[a,t]})$. Then $s(a)=0$, $s(b)=L$, and $s$ is nondecreasing by [step 1.1]; moreover $s(w)-s(v)=L(\gamma {\mathrm{\mid }}_{[v,w]})$ for all $a\le v\le w\le b$, by additivity at the initial point [step 1.1].

(Equicontinuity of bounded arc-length families.) Let ${\gamma }_k\text{\hspace{0.17em}⁣}:[0,1]\to X$ be arc-length parametrized with $L({\gamma }_k)\le M<\mathrm{\infty }$. For $0\le s\le t\le 1$ the chord bound [step 1.1] and the definition of arc-length parametrized give $d({\gamma }_k(s),{\gamma }_k(t))\le L({\gamma }_k{\mathrm{\mid }}_{[s,t]})=(t-s)L({\gamma }_k)\le M(t-s)$, so every ${\gamma }_k$ is $M$-Lipschitz. If $M=0$ then every ${\gamma }_k$ is constant by separation [F1] and the family is uniformly equicontinuous with any $\delta >0$. If $M>0$ and $\epsilon >0$ is real, take $\delta :=\epsilon \mathrm{/}M>0$; then $d({\gamma }_k(s),{\gamma }_k(t))\le M\mathrm{\mid }s-t\mathrm{\mid }<\epsilon$ for all $k$ and all $s,t$ with $\mathrm{\mid }s-t\mathrm{\mid }<\delta$, so the family is uniformly equicontinuous, hence equicontinuous [F10].

($s$ is continuous.) Fix $t\in [a,b)$ and a real $\delta >0$; put $\epsilon :=\delta \mathrm{/}2$. Since $L$ is the least upper bound of the polygonal sums, choose a partition $P$ of $[a,b]$ with polygonal sum $S(P)>L-\epsilon$ [F3], and insert $t$ into $P$ if absent (the sum only increases [F1]); write $A$ and $B$ for the sums of the parts of $P$ on $[a,t]$ and on $[t,b]$, so that $A+B=S(P)$, and let $t^{+}$ be the successor of $t$ in $P$ when $t<b$. By continuity of $\gamma$ at $t$ [F9] choose $\eta \in (0,t^{+}-t]$ with $d(\gamma (t),\gamma (t^{\mathrm{\prime }}))<\epsilon$ for $\mathrm{\mid }{t}^{\mathrm{\prime }}-t\mathrm{\mid }<\eta$; I claim $s(t+h)-s(t)\le \delta$ for every $h\in (0,\eta )$. Let $R$ be a partition of $[t,t+h]$; then the union of the parts of $P$ on $[a,t]$, of $R$, and of the part of $P$ on $[t^{+},b]$ is a partition of $[a,b]$ whose polygonal sum is $A+\mathrm{S}\mathrm{u}\mathrm{m}(R)+d(\gamma (t+h),\gamma (t^{+}))+(B-d(\gamma (t),\gamma (t^{+})))\le L$, so $\mathrm{S}\mathrm{u}\mathrm{m}(R)\le L-S(P)+d(\gamma (t),\gamma (t^{+}))-d(\gamma (t+h),\gamma (t^{+}))\le L-S(P)+d(\gamma (t),\gamma (t+h))<\epsilon +\epsilon =2\epsilon$ by the reverse triangle inequality [F2]. Taking the supremum over $R$ gives $s(t+h)-s(t)=L(\gamma {\mathrm{\mid }}_{[t,t+h]})\le 2\epsilon =\delta$ [step 2.1]. The same argument applied to partitions of $[t-h,t]$ gives left-continuity at every $t\in (a,b]$; hence $s$ is continuous.

($s$ is surjective.) The interval $[a,b]$ is connected [F8] and $s\text{\hspace{0.17em}⁣}:[a,b]\to \mathbb{R}$ is continuous [step 3.1] with $s(a)=0$ and $s(b)=L$ [step 2.1]; by the intermediate value theorem [F8], for every real $r$ with $0\le r\le L$ there is $t\in [a,b]$ with $s(t)=r$.

(Factorisation through $s$.) If $a\le u\le v\le b$ satisfy $s(u)=s(v)$, then $L(\gamma {\mathrm{\mid }}_{[u,v]})=s(v)-s(u)=0$ [step 2.1], so $d(\gamma (u),\gamma (v))\le 0$ by the chord bound [step 1.1] and hence $\gamma (u)=\gamma (v)$ by separation [F1]. Since $s$ is surjective [step 4.1], there is therefore a well-defined and unique map $\bar{\gamma }\text{\hspace{0.17em}⁣}:[0,L]\to X$ with $\gamma =\bar{\gamma }\circ s$, namely $\bar{\gamma }(r):=\gamma (t)$ for any $t$ with $s(t)=r$. If $L=0$ then $s$ vanishes identically and the same argument with $u=a$, $v=b$ shows that $\gamma$ is constant; then $[0,L]=\{0\}$ and $\bar{\gamma }$ is that constant.

($\bar{\gamma }$ is $1$-Lipschitz and has unit speed.) Given $0\le r\le q\le L$, choose by [step 4.1] points $u,v\in [a,b]$ with $s(u)=r$ and $s(v)=q$, and relabel so that $u\le v$; then $d(\bar{\gamma }(r),\bar{\gamma }(q))=d(\gamma (u),\gamma (v))\le L(\gamma {\mathrm{\mid }}_{[u,v]})=s(v)-s(u)=q-r$ by the chord bound [step 1.1] and [step 2.1], so $\bar{\gamma }$ is $1$-Lipschitz. If $r=q$, the singleton-interval convention gives $L(\bar{\gamma }{\mathrm{\mid }}_{[r,r]})=0=q-r$; hence assume $r<q$ for the length identity. Define, for each real $\rho$ with $0\le \rho \le L$, the number $u_{\rho }:=\sup \{t\in [a,b]:s(t)\le \rho \}$, which exists by the least-upper-bound property [F8] and satisfies $s(u_{\rho })=\rho$: indeed $s(u_{\rho })\le \rho$ because points of the set approach $u_{\rho }$ from below and $s$ is continuous [step 3.1], while if $u_{\rho }<b$ then every $t>u_{\rho }$ has $s(t)>\rho$ and continuity gives $s(u_{\rho })\ge \rho$, and if $u_{\rho }=b$ then $s(u_{\rho })=L\le \rho \le L$. Also $u_{\rho }\le u_{{\rho }^{\mathrm{\prime }}}$ for $\rho \le {\rho }^{\mathrm{\prime }}$, and $\bar{\gamma }(\rho )=\gamma (u_{\rho })$ [step 5.1]. Now let $r={\rho }_0<\cdots <{\rho }_N=q$ be a partition of $[r,q]$; its polygonal sum for $\bar{\gamma }$ is ${\sum }_{i}d(\gamma (u_{{\rho }_{i-1}}),\gamma (u_{{\rho }_i}))\le {\sum }_{i}L(\gamma {\mathrm{\mid }}_{[u_{{\rho }_{i-1}},u_{{\rho }_i}]})={\sum }_i({\rho }_i-{\rho }_{i-1})=q-r$ by the chord bound [step 1.1] and [step 2.1], so $L(\bar{\gamma }{\mathrm{\mid }}_{[r,q]})\le q-r$. Conversely let $v_0<\cdots <v_N$ be a partition of $[u_r,u_q]$; its polygonal sum for $\gamma$ is ${\sum }_{i}d(\bar{\gamma }(s(v_{i-1})),\bar{\gamma }(s(v_i)))$, the points $s(v_i)$ form a nondecreasing sequence from $r=s(v_0)$ to $q=s(v_N)$, and after deleting repetitions this is a partition of $[r,q]$ whose polygonal sum for $\bar{\gamma }$ is the same number; hence every polygonal sum of $\gamma {\mathrm{\mid }}_{[u_r,u_q]}$ is at most $L(\bar{\gamma }{\mathrm{\mid }}_{[r,q]})$, and $q-r=L(\gamma {\mathrm{\mid }}_{[u_r,u_q]})\le L(\bar{\gamma }{\mathrm{\mid }}_{[r,q]})$ [step 2.1]. Therefore $L(\bar{\gamma }{\mathrm{\mid }}_{[r,q]})=q-r$.

(Application.) Let $X$ carry the chain metric $d$ of an isometric polyhedral gluing with (H1)-(H3); by [F11] this $d$ is a metric on $X$, so clauses (i)-(iii), whose statements and proofs mention only the metric space $(X,d)$, hold verbatim for paths in $(X,d)$; in particular the constants $a<b$ are arbitrary reals and no hypothesis beyond the metric axioms was used.