Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas

Gram realisation. By [F1], $C=L{L}^T$ with $L$ invertible, so the rows $u_i$ of $L$ satisfy $u_i\cdot u_j=(L{L}^T{)}_{ij}=c_{ij}$, in particular $\mathrm{\mid }{u}_i{\mathrm{\mid }}^2=c_{ii}=1$, and they are linearly independent because $\det L\mathrm{\ne }0$. If $(u_i^{\mathrm{\prime }})$ is any family of unit vectors of a Euclidean space with $u_i^{\mathrm{\prime }}\cdot u_j^{\mathrm{\prime }}=c_{ij}$ and ${\sum }_i{\lambda }_i{u}_i=0$, then $\mid {\sum }_i{\lambda }_i{u}_i^{\mathrm{\prime }}{\mid }^2={\lambda }^{T}C\lambda =\mid {\sum }_i{\lambda }_i{u}_i{\mid }^2=0$, so ${\sum }_i{\lambda }_i{u}_i^{\mathrm{\prime }}=0$; the two families therefore obey the same linear relations, the map $T$ sending $u_i$ to $u_i^{\mathrm{\prime }}$ is a well-defined linear isomorphism of their spans, and it preserves inner products because these are given by the Gram matrices on both sides; $T$ extends to a linear isometry of ${\mathbb{R}}^{n+1}$ onto the span of the $u_i^{\mathrm{\prime }}$, which carries $K(C)$ onto the cone generated by the $u_i^{\mathrm{\prime }}$ and $\mathrm{\Sigma }(C)$ onto its unit section.

The two tangent-cone descriptions. Let $C$ be given by the nonconstant inequalities ${\mathrm{\ell }}_j\ge 0$ after the discarding prescribed in the definition, and let $p$ lie in the relative interior of the face $F$, with $I(F)$ the set of $j$ with ${\mathrm{\ell }}_j(p)=0$ and $n_j=\mathrm{\nabla }{\mathrm{\ell }}_j\mathrm{/}\mathrm{\mid }\mathrm{\nabla }{\mathrm{\ell }}_j\mathrm{\mid }$ inward. If $\xi$ satisfies $⟨\xi ,n_j⟩\ge 0$ for all $j\in I(F)$, then for small $\epsilon >0$ the point $p+\epsilon \xi$ satisfies ${\mathrm{\ell }}_j(p+\epsilon \xi )={\mathrm{\ell }}_j(p)+\epsilon ⟨\mathrm{\nabla }{\mathrm{\ell }}_j,\xi ⟩=\epsilon \mathrm{\mid }\mathrm{\nabla }{\mathrm{\ell }}_j\mathrm{\mid }⟨n_j,\xi ⟩\ge 0$ for $j\in I(F)$ and ${\mathrm{\ell }}_j(p+\epsilon \xi )>0$ for $j\mathrm{\notin }I(F)$ by continuity and ${\mathrm{\ell }}_j(p)>0$, so $p+\epsilon \xi \in C$ and $\xi$ is a limit of vectors $\lambda (x-p)$ with $x\in C$. Conversely, for $x\in C$ and $j\in I(F)$ the affine identity ${\mathrm{\ell }}_j(x)={\mathrm{\ell }}_j(p)+⟨x-p,\mathrm{\nabla }{\mathrm{\ell }}_j⟩$ gives $⟨x-p,n_j⟩={\mathrm{\ell }}_j(x)\mathrm{/}\mathrm{\mid }\mathrm{\nabla }{\mathrm{\ell }}_j\mathrm{\mid }\ge 0$, so every $\lambda (x-p)$ with $\lambda \ge 0$ lies in the closed cone $\{\xi :⟨\xi ,n_j⟩\ge 0(j\in I(F))\}$; the two cones agree, and since an affine function nonnegative on $F$ vanishes at a relative interior point exactly when it vanishes throughout $F$, the active set does not depend on $p$, and the tangent cone $T_{F}C$ does not depend on the chosen point of the relative interior. The common kernel of the active gradients is $U(F)$: it contains $U(F)$ because the active functions vanish on $F$; conversely, for a vector in that kernel, sufficiently small displacements $p\pm \epsilon \xi$ remain in $C$ and satisfy all active equalities, hence belong to $F$ by [F5], so $\xi \in U(F)$.

The angular distance is a metric. For unit vectors $\xi ,\eta$ the number $d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\xi ,\eta )=\arccos ⟨\xi ,\eta ⟩\in [0,\pi ]$ is well defined by [F2], is symmetric, vanishes exactly when $⟨\xi ,\eta ⟩=1$, i.e. $\xi =\eta$, and is at most $\pi$. If $\xi ,\zeta ,\eta$ are unit, $\alpha :=d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\xi ,\zeta )$ and $\beta :=d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\zeta ,\eta )$ satisfy $\alpha +\beta \le \pi$, write (trivially if $\alpha =0$ or $\beta =0$) $\xi =\cos \alpha \zeta +\sin \alpha a$ and $\eta =\cos \beta \zeta +\sin \beta b$ with $a,b$ unit and orthogonal to $\zeta$ by [F2]; then $⟨\xi ,\eta ⟩=\cos \alpha \cos \beta +\sin \alpha \sin \beta ⟨a,b⟩\ge \cos \alpha \cos \beta -\sin \alpha \sin \beta =\cos (\alpha +\beta )$ by [F2] and [F3], so $d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\xi ,\eta )\le \alpha +\beta$ because $\arccos$ is the inverse of the strictly decreasing $\cos$ on $[0,\pi ]$; if $\alpha +\beta >\pi$ the same inequality is trivial. So $d_{\mathrm{a}\mathrm{n}\mathrm{g}}$ is a metric of diameter at most $\pi$ on every set of unit vectors.

The hemisphere functional. Since $(u_i)$ is a basis of ${\mathbb{R}}^{n+1}$ by step 1.1, there is a unique linear functional $\phi$ with $\phi (u_i)=1$ for all $i$, and $\phi \left({\sum }_i{\lambda }_i{u}_i\right)={\sum }_i{\lambda }_i$ for all $\lambda$. If $x={\sum }_i{\lambda }_i{u}_i\in K(C)\setminus \{0\}$ with ${\lambda }_i\ge 0$, then some ${\lambda }_i>0$, so $\phi (x)={\sum }_i{\lambda }_i>0$; hence $\mathrm{\Sigma }(C)\subseteq \{x\in S^n:\phi (x)>0\}$ and $0\mathrm{\notin }\mathrm{\Delta }(C):=\{x\in K(C):\phi (x)=1\}$, which is the set of convex combinations ${\sum }_i{\lambda }_i{u}_i$ with ${\lambda }_i\ge 0$, ${\sum }_i{\lambda }_i=1$. In standard coordinates $\phi (x)=w\cdot x$ for a unique nonzero vector $w$; [F2] gives $\mathrm{\mid }\phi (x)\mathrm{\mid }\le \mathrm{\mid }w\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }$, with equality at $x=w\mathrm{/}\mathrm{\mid }w\mathrm{\mid }$, so $\parallel \phi \parallel :=\mathrm{\mid }w\mathrm{\mid }$ is the finite operator norm.

Schur complements. Since $\mathrm{\mid }{c}_{i0}\mathrm{\mid }<1$ for $i\ge 1$ by [F2] and the linear independence of $u_i,u_0$ of step 1.1, the vectors $v_i:=(u_i-c_{i0}{u}_0)\mathrm{/}\sqrt{1-c_{i0}^2}$ are well defined; they satisfy $v_i\cdot u_0=0$ and $\mathrm{\mid }{v}_i\mathrm{\mid }=1$, and expanding $v_i\cdot v_j$ gives $v_i\cdot v_j=(c_{ij}-c_{i0}{c}_{j0})\mathrm{/}\sqrt{(1-c_{i0}^2)(1-c_{j0}^2)}=c_{ij}^{\mathrm{l}\mathrm{k}}$, in particular $c_{ii}^{\mathrm{l}\mathrm{k}}=1$. Writing $w:=(-{\sum }_j{c}_{0j}{x}_j,x_1,\dots ,x_n)$ for $x\in {\mathbb{R}}^n\setminus \{0\}$, the vector $w$ is nonzero and, with $c_{00}=1$, $w^{T}Cw={\sum }_{i,j\ge 1}{c}_{ij}{x}_i{x}_j-({\sum }_j{c}_{0j}{x}_j{)}^2>0$; the quadratic form on the right is that of the unnormalised Schur complement $Z:=(c_{ij}-c_{i0}{c}_{j0}{)}_{i,j\ge 1}$, so $Z$ is positive definite, and $C^{\mathrm{l}\mathrm{k}}=S^{-1\mathrm{/}2}Z{S}^{-1\mathrm{/}2}$ with the positive diagonal $S=\mathrm{diag}(1-c_{i0}^2)$ is positive definite as well.

The intrinsic path metric of a link. For a continuous path $\gamma$ in $V$ write $\mathrm{\ell }(\gamma )$ for the supremum of ${\sum }_i\mathrm{\mid }\gamma (t_{i+1})-\gamma (t_i)\mathrm{\mid }$ over finite partitions of its parameter interval, and for $\xi ,\eta \in {\mathrm{Lk}}_C(F)$ let $d_{\mathrm{i}\mathrm{n}\mathrm{t}}(\xi ,\eta )$ be the infimum of $\mathrm{\ell }(\gamma )$ over continuous paths in ${\mathrm{Lk}}_C(F)$ from $\xi$ to $\eta$. Let $\theta =d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\xi ,\eta )$. The cone $N_{F}C$ is pointed: a vector and its negative lying in it satisfy every active facet equality, hence lie in the direction space of the affine hull of $F$, namely $U(F)$; orthogonality to $U(F)$ then forces the vector to be zero. Thus distinct unit directions are never antipodal and $0<\theta <\pi$. If $\theta >0$, pick a unit $w\in {\xi }^{\perp }$ with $\eta =\cos \theta \xi +\sin \theta w$ and put $\gamma (t):=\cos t\xi +\sin tw$ for $t\in [0,\theta ]$; then $\gamma ([0,\theta ])\subseteq N_{F}C\cap S(V)$ because $\gamma (t)=\frac{\sin (\theta -t)}{\sin \theta }\xi +\frac{\sin t}{\sin \theta }\eta$ is a nonnegative combination of $\xi ,\eta \in N_{F}C$ for $0\le t\le \theta$, and for every partition ${\sum }_i\mathrm{\mid }\gamma (t_{i+1})-\gamma (t_i)\mathrm{\mid }={\sum }_{i}2\sin \frac{t_{i+1}-t_i}{2}\le {\sum }_i(t_{i+1}-t_i)=\theta$ while the uniform partition into $m$ parts has sum $2m\sin \frac{\theta }{2m}\to \theta$ as $m\to \mathrm{\infty }$ by [F3]; hence $\mathrm{\ell }(\gamma )=\theta$ and $d_{\mathrm{i}\mathrm{n}\mathrm{t}}(\xi ,\eta )\le \theta$. Conversely let $\gamma :[a,b]\to N_{F}C\cap S(V)$ be any continuous path from $\xi$ to $\eta$ and let $f(t):=d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\xi ,\gamma (t))$, a continuous function with $f(a)=0$ and $f(b)=\theta$; for each $m\ge 1$ successively applying [F4] on $[t_{i-1},b]$ gives $a=t_0\le t_1\le \cdots \le t_m=b$ with $f(t_i)=i\theta \mathrm{/}m$, and then ${\sum }_i\mathrm{\mid }\gamma (t_i)-\gamma (t_{i-1})\mathrm{\mid }={\sum }_{i}2\sin \frac{{\phi }_i}{2}$ with ${\phi }_i:=d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\gamma (t_{i-1}),\gamma (t_i))\ge \mathrm{\mid }f(t_i)-f(t_{i-1})\mathrm{\mid }=\theta \mathrm{/}m$ by step 1.3 and [F2]; since ${\phi }_i\in [0,\pi ]$ and $\sin$ is strictly increasing on $[0,\pi \mathrm{/}2]$, each term is at least $2\sin \frac{\theta }{2m}$, so $\mathrm{\ell }(\gamma )\ge 2m\sin \frac{\theta }{2m}\to \theta$ and $d_{\mathrm{i}\mathrm{n}\mathrm{t}}(\xi ,\eta )\ge \theta$; for $\theta =0$ both numbers are $0$. Hence $d_{\mathrm{i}\mathrm{n}\mathrm{t}}=d_{\mathrm{a}\mathrm{n}\mathrm{g}}$ on ${\mathrm{Lk}}_C(F)$, the link has diameter at most $\pi$, and $d_{\mathrm{a}\mathrm{n}\mathrm{g}}$ is the intrinsic path metric of the round unit sphere restricted to the link.

Links of gluings. Let $X$ be an isometric polyhedral gluing with cells $C_p$ and let $F$ be a face. By step 1.2 the tangent cone $T_F{C}_p$ and the angular link ${\mathrm{Lk}}_{C_p}(F)$ depend only on $F$, and for $p\le q$ the isometry $h_{p,q}:C_p\to C_q$ maps $F$ onto $F$, so its linear part maps $T_F{C}_p$ isometrically into $T_F{C}_q$ and maps its normal section onto the corresponding link face that is compatible with the cocycle condition; the intersection condition makes the intersections of the resulting images correspond to the common faces, so the cell links glue to a well-defined quotient ${\mathrm{Lk}}_X(F)$, and replacing the point of $F$ changes nothing by step 1.2. The normal sections in cofaces $G>F$ inherit their polyhedral face incidences; when $X$ is simplicial these correspond to the simplices $G\setminus F$ of its combinatorial link. A general polyhedral face link need not itself be simplicial.

Radial coordinates. For $x\in \mathrm{\Delta }(C)$ the vector $\rho (x)=x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ lies in $K(C)\cap S^n=\mathrm{\Sigma }(C)$, and for $y\in \mathrm{\Sigma }(C)$, written $y={\sum }_i{\lambda }_i{u}_i$ with ${\lambda }_i\ge 0$, the number $\phi (y)={\sum }_i{\lambda }_i>0$ is positive, the point $y\mathrm{/}\phi (y)={\sum }_i({\lambda }_i\mathrm{/}\phi (y))u_i$ lies in $\mathrm{\Delta }(C)$ and normalises back to $y$; so $y\mapsto y\mathrm{/}\phi (y)$ is an inverse of $\rho$ and $\rho$ is a bijection $\mathrm{\Delta }(C)\to \mathrm{\Sigma }(C)$. It is injective because $x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }=x^{\mathrm{\prime }}\mathrm{/}\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }$ gives $x=a{x}^{\mathrm{\prime }}$ with $a=\mathrm{\mid }x\mathrm{\mid }\mathrm{/}\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }=\phi (x)\mathrm{/}\phi (x^{\mathrm{\prime }})=1$. The coefficients ${\lambda }_i$ of a point of $\mathrm{\Delta }(C)$ in the basis $(u_i)$ are unique, so every point of $\mathrm{\Sigma }(C)$ has unique barycentric ray coordinates ${\lambda }_i\ge 0$ with ${\sum }_i{\lambda }_i=1$.

Iterated links and face compatibility. Let $A$ be a nonempty proper set of vertices (if it is the full set, the normal link is empty), put $P_A:=\mathrm{span}\{u_a:a\in A\}$, and write $b_i=(c_{ai}{)}_{a\in A}$. The orthogonal projection of $u_i$ onto $P_A^{\perp }$ is $w_i=u_i-{\sum }_{a\in A}(C_A^{-1}{b}_i{)}_a{u}_a$: pairing with each $u_a$ gives zero, while the subtracted vector lies in $P_A$. Consequently $⟨w_i,w_j⟩=c_{ij}-b_i^T{C}_A^{-1}{b}_j$, the Schur complement $Z$ of $C_A$, and $Z$ is positive definite since for nonzero $x$ on the complementary indices, $(-C_A^{-1}Bx,x{)}^{T}C(-C_A^{-1}Bx,x)=x^{T}Zx>0$, where $B=(c_{ai})$. In particular all $w_i$ are nonzero; normalising them gives the diagonal-$1$ Gram matrix $z_{ij}\mathrm{/}\sqrt{z_{ii}{z}_{jj}}$. These projected rays describe the normal link: at an interior point of the face, coefficients of the face vertices can vary in both directions, whereas coefficients of the remaining vertices must be nonnegative; quotienting out the face span therefore leaves exactly the cone generated by the $w_i$. For iteration, after removing a vertex, project the remaining vectors onto the perpendicular of its residual projected direction, not onto the perpendicular of the original vector. For a relative interior point $p$ of the spherical face, $p$ is a positive combination of its face vertices. A tangent vector $v$ with $v\perp p$ points into the simplex exactly when its coefficients outside $A$ are nonnegative: for small $\epsilon >0$, the vector $p+\epsilon v$ retains positive coefficients in $A$, and its normalisation has initial direction $v$, as $(p+\epsilon v)\mathrm{/}\mathrm{\mid }p+\epsilon v\mathrm{\mid }=p+\epsilon v+O({\epsilon }^2)$. Removing the tangent face directions $P_A\cap p^{\perp }$ leaves $P_A^{\perp }$ and the projected rays $w_i$, proving the asserted geometric link identification. Those residual directions obtained from an ordering of $A$ are orthogonal and span $P_A$ (each new residual is orthogonal to the preceding span and is nonzero by independence); their successive perpendicular projections therefore equal the single orthogonal projection onto $P_A^{\perp }$. Intermediate normalisations only rescale rays by positive factors, so the final normalised Gram matrix is the same in every order. Restricting to a subface retains the corresponding subset of remaining rays, while linking a further face enlarges $A$ and repeats this operation.

Lipschitz estimates and the chord bound. $\mathrm{\Delta }(C)$ is a nonempty compact convex cell and avoids $0$ by step 2.1 and [F5]. The reverse norm triangle inequality makes $z\mapsto \mathrm{\mid }z\mathrm{\mid }$ continuous, so [F9] gives an attained minimum $m_0:={\min }_{z\in \mathrm{\Delta }(C)}\mathrm{\mid }z\mathrm{\mid }>0$, and for $x,y\in \mathrm{\Delta }(C)$ the estimate $\mid \frac{x}{\mathrm{\mid }x\mathrm{\mid }}-\frac{y}{\mathrm{\mid }y\mathrm{\mid }}\mid \le \frac{\mathrm{\mid }x-y\mathrm{\mid }}{\mathrm{\mid }x\mathrm{\mid }}+\frac{\mid \mathrm{\mid }x\mathrm{\mid }-\mathrm{\mid }y\mathrm{\mid }\mid }{\mathrm{\mid }x\mathrm{\mid }}\le \frac{2\mathrm{\mid }x-y\mathrm{\mid }}{m_0}$ holds. On $\mathrm{\Sigma }(C)$ one has $\mathrm{\mid }y\mathrm{\mid }=1$ and $\phi (y)\ge 1$ (because $y=z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ with $z\in \mathrm{\Delta }(C)$ and $\mathrm{\mid }z\mathrm{\mid }\le 1$), so writing $y\mathrm{/}\phi (y)-y^{\mathrm{\prime }}\mathrm{/}\phi (y^{\mathrm{\prime }})=\frac{(y-y^{\mathrm{\prime }})}{\phi (y)}+\frac{y^{\mathrm{\prime }}(\phi (y^{\mathrm{\prime }})-\phi (y))}{\phi (y)\phi (y^{\mathrm{\prime }})}$ gives $\mid {\rho }^{-1}(y)-{\rho }^{-1}(y^{\mathrm{\prime }})\mid \le \mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }+\parallel \phi \parallel \mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }$. For unit vectors $\xi ,\eta$ put $\theta :=d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\xi ,\eta )=\arccos ⟨\xi ,\eta ⟩$; then $\mathrm{\mid }\xi -\eta {\mathrm{\mid }}^2=2-2⟨\xi ,\eta ⟩=2-2\cos \theta =4{\sin }^2(\theta \mathrm{/}2)$ by [F3], so the chord length is $\text{chord}=2\sin (\theta \mathrm{/}2)\le \theta =\text{round}$, the last step because $\sin u\le u$ for $u\ge 0$ by [F3]. For the reverse comparison note first that $\pi >2$: for $0<u\le \pi \mathrm{/}2$ the mean value theorem gives $\sin u=u\cos \xi$ with $\xi \in (0,u)$, and $\cos \xi <\cos 0=1$ because $\cos$ is strictly decreasing on $[0,\pi ]$ [F3], so $\sin u<u$ and $1=\sin (\pi \mathrm{/}2)<\pi \mathrm{/}2$; the restriction of $\cos$ to $[0,\pi ]$ is strictly decreasing with $\cos 0=1$ and $\cos (\pi \mathrm{/}2)=0$ by [F2] and [F3], so $2\mathrm{/}\pi \in (0,1)$ and $u_{\ast }:=\arccos (2\mathrm{/}\pi )$ is the unique point of $[0,\pi ]$ with $\cos u_{\ast }=2\mathrm{/}\pi$, lying in $(0,\pi \mathrm{/}2)$; the function $f(u):=\sin u-2u\mathrm{/}\pi$ therefore has $f^{\mathrm{\prime }}(u)=\cos u-2\mathrm{/}\pi$ by [F3], positive on $[0,u_{\ast })$ and negative on $(u_{\ast },\pi \mathrm{/}2]$. The mean value theorem [F3] gives $f(u)\ge f(0)=0$ on $[0,u_{\ast }]$ and $f(u)\ge f(\pi \mathrm{/}2)=0$ on $[u_{\ast },\pi \mathrm{/}2]$ (with $f(\pi \mathrm{/}2)=\sin (\pi \mathrm{/}2)-1=0$), that is, $\sin u\ge 2u\mathrm{/}\pi$ on $[0,\pi \mathrm{/}2]$. Applied to $u=\theta \mathrm{/}2\le \pi \mathrm{/}2$ this gives $\theta \le (\pi \mathrm{/}2)2\sin (\theta \mathrm{/}2)=(\pi \mathrm{/}2)\text{chord}$, so the chord and round distances satisfy $\text{chord}\le \text{round}\le \frac{\pi }{2}\text{chord}$.

The finite spherical complex and the descent map. By steps 1.1 and 3.2 the identifications $\mathrm{\Sigma }(C_{\tau })\to \mathrm{\Sigma }(C_{\sigma })$ for $\tau \subseteq \sigma$ are well-defined isometries onto faces, so $X=\mathrm{\mid }K{\mathrm{\mid }}_C$ is a well-defined quotient of the finite disjoint union of the compact simplices $\mathrm{\Sigma }(C_{\sigma })$ by finitely many face identifications. Let $Y$ be the corresponding isometric polyhedral gluing with cells $\mathrm{\Delta }(C_{\sigma })$; the cell maps ${\mathrm{\Phi }}_{\sigma }:\mathrm{\Sigma }(C_{\sigma })\to \mathrm{\Delta }(C_{\sigma })$, ${\mathrm{\Phi }}_{\sigma }(y):=y\mathrm{/}{\phi }_{\sigma }(y)$, agree on identified faces, because ${\phi }_{\sigma }$ takes the value $1$ on all vertices in $\sigma$ and hence restricts to ${\phi }_{\tau }$; so the ${\mathrm{\Phi }}_{\sigma }$ descend to a bijection $\mathrm{\Phi }:X\to Y$.

The link of a face of a finite spherical complex. Let $X=\mathrm{\mid }K{\mathrm{\mid }}_C$ be a finite spherical complex and let $F$ be the face corresponding to a nonempty simplex $\sigma$. The cells of ${\mathrm{Lk}}_X(F)$ are the links ${\mathrm{Lk}}_{C_{\tau }}(F)$ of the spherical simplices containing $F$; by step 3.2 each is again a spherical simplex, with Gram matrix obtained by iterating the Schur complements of step 2.2 over the vertices of $\sigma$, and these matrices are positive definite of diagonal $1$ and compatible with faces, because passing to a smaller coface retains a subset of the projected rays, while linking a further face enlarges the projected-out span; the identifications are those induced by $K$ and are isometric by step 2.4. Hence ${\mathrm{Lk}}_X(F)$ is again presented as a finite spherical complex, and clauses (i)-(v) apply to it.

Bi-Lipschitz equivalence of the model and its spherical simplex. With $m_0={\min }_{z\in \mathrm{\Delta }(C)}\mathrm{\mid }z\mathrm{\mid }>0$ and the operator norm $\parallel \phi \parallel$, steps 3.1 and 4.1 give for $y=\rho (x)$, $y^{\mathrm{\prime }}=\rho (x^{\mathrm{\prime }})$ the bounds $d_{\mathrm{a}\mathrm{n}\mathrm{g}}(y,y^{\mathrm{\prime }})\le \frac{\pi }{2}\mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }\le \frac{\pi }{m_0}\mathrm{\mid }x-x^{\mathrm{\prime }}\mathrm{\mid }$ and $\mathrm{\mid }x-x^{\mathrm{\prime }}\mathrm{\mid }\le (1+\parallel \phi \parallel )\mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid }\le (1+\parallel \phi \parallel )d_{\mathrm{a}\mathrm{n}\mathrm{g}}(y,y^{\mathrm{\prime }})$, so $\rho$ is a bi-Lipschitz equivalence between the Euclidean metric of $\mathrm{\Delta }(C)$ and the round metric of $\mathrm{\Sigma }(C)$ with the constant $L:=\max \{\pi \mathrm{/}{m}_0,1+\parallel \phi \parallel \}$.

The radial map is bi-Lipschitz for the chain metrics. If $K$ has no nonempty simplex then $X=Y=\mathrm{\varnothing }$ and there are no distance estimates to check. Otherwise, by step 5.1 each ${\mathrm{\Phi }}_{\sigma }$ is bi-Lipschitz with constant $L_{\sigma }=\max \{\pi \mathrm{/}{m}_0^{\sigma },1+\parallel {\phi }_{\sigma }\parallel \}$, and the finitely many simplices of $K$ give finite maxima $L^{\mathrm{\prime }}:={\max }_{\sigma }(1+\parallel {\phi }_{\sigma }\parallel )$ and $L^{\mathrm{\prime }\mathrm{\prime }}:={\max }_{\sigma }\pi \mathrm{/}{m}_0^{\sigma }$. Every chain in $X$ maps under $\mathrm{\Phi }$ to a chain in $Y$ whose one-step Euclidean distances are at most $L^{\mathrm{\prime }}$ times the angular distances of the original chain, so $d_Y(\mathrm{\Phi }x,\mathrm{\Phi }{x}^{\mathrm{\prime }})\le L^{\mathrm{\prime }}{d}_X(x,x^{\mathrm{\prime }})$; likewise every chain in $Y$ is the image of a chain in $X$ with angular steps at most $L^{\mathrm{\prime }\mathrm{\prime }}$ times the Euclidean steps, so $d_X(x,x^{\mathrm{\prime }})\le L^{\mathrm{\prime }\mathrm{\prime }}{d}_Y(\mathrm{\Phi }x,\mathrm{\Phi }{x}^{\mathrm{\prime }})$. Hence $\mathrm{\Phi }$ is a bi-Lipschitz equivalence of the chain metrics.

Component metrics and topology. The classes under being joined by a chain are finite in number. Each is a union of whole cells: any two points of one cell are joined by a single chain step. The trace of a class on every cell is therefore the entire cell or empty, so every class is weakly open and closed; each class is path connected by cell arcs, hence is a connected component. The corresponding component of $Y$ satisfies (H1)-(H3), so [F7] supplies its metric, weak topology, properness and completeness; step 6.1 transfers these properties to the component of $X$. Each component is compact as a quotient of finitely many compact cells. Its chain metric is a length metric: a chain can be replaced by cell arcs, whose $d$-length is at most its chain length because $d\le d_{\mathrm{a}\mathrm{n}\mathrm{g}}$ on a cell, while every path has length at least the distance of its endpoints. Cellwise radial homeomorphisms descend to a homeomorphism for the quotient (weak) topologies, and the components are open and closed, so $X$ and $Y$ are compact finite disjoint unions of the component metric spaces. Distances between distinct components remain auxiliary infinities; no real-valued global chain metric is asserted. The empty case is vacuous.

Minimizing geodesics (AC). Assume AC, fix $x,y$ in one component and put $R=d(x,y)$. AC selects, for each $n\ge 1$, a chain of length $L_n<R+1\mathrm{/}n$; the infimum definition gives $R\le L_n$. Replace its steps by cell arcs, traversed at angular speed $L_n$ on $[0,1]$ (use the constant path for $L_n=0$). Each arc lies in its cell by the nonnegative-combination argument of step 2.3 applied to $K(C_{\sigma })$, and the triangle inequality gives $d({\gamma }_n(s),{\gamma }_n(t))\le L_n\mathrm{\mid }s-t\mathrm{\mid }$, hence $R\le \mathrm{\ell }({\gamma }_n)\le L_n$. The paths are equicontinuous and pointwise bounded in the compact proper component of step 7.1. The domain $[0,1]$ is compact as a nonempty bounded one-dimensional polyhedral cell by [F5]; with AC it satisfies [F8], giving a uniformly convergent subsequence with continuous limit $\gamma$ and endpoints $x,y$. Since $R\le \mathrm{\ell }({\gamma }_n)\le L_n\to R$, lower semicontinuity [F7] gives $\mathrm{\ell }(\gamma )\le R$, while the endpoint partition gives the reverse bound. Reparametrise by arc length using [F7]. For $0\le u\le v\le R$, the resulting path $\bar{\gamma }$ is $1$-Lipschitz, and $R=d(x,y)\le u+d(\bar{\gamma }(u),\bar{\gamma }(v))+R-v$ forces $d(\bar{\gamma }(u),\bar{\gamma }(v))\ge v-u$; thus equality holds and $\bar{\gamma }$ is a minimizing geodesic. AC is used for the sequence selection and for Ascoli.

The $+\mathrm{\infty }$ convention. Let $F$ be a face of a finite spherical complex $X$. By step 4.3 the angular link ${\mathrm{Lk}}_X(F)$ is a finite spherical complex, so step 7.1 applies to it: being joined by a chain of directions inside common cells is reflexive (singleton chains), symmetric (reverse a chain) and transitive (concatenate chains), hence an equivalence relation on ${\mathrm{Lk}}_X(F)$; the infimum $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}(x,y)$ is finite whenever a chain from $x$ to $y$ exists and is a metric on each class by step 7.1, while for $x,y$ in different classes no chain joins them and $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}(x,y)=+\mathrm{\infty }$. This infinite value is auxiliary notation, not an ordinary distance of the metric-space definition, and it is replaced by $\pi$ in the truncated metric of The angular path metric, the Euclidean cone and spherical joins.