Face coherence, global hat coordinates and a uniform star radius

Every nonempty face $p$ has finitely many vertices and at least one, and its barycentre $b_p$ lies in the relative interior of $C_p$. By [F2] the cell $C_p$ has finitely many faces, and a $0$-dimensional face is a singleton by [F1]. If $\dim C_p=0$ the cell is its own only face. If $\dim C_p\ge 1$, then not all defining inequalities of $C_p$ are constant on its affine span, since otherwise $C_p$ would be that whole affine space, which is unbounded in positive dimension, or empty; so some defining inequality $\mathrm{\ell }$ is nonconstant on $C_p$ and satisfies $\mathrm{\ell }\ge 0$ there. In orthonormal affine coordinates $\mathrm{\ell }=c+{\sum }_{i=1}^n{a}_i{x}_i$ satisfies $\mathrm{\mid }\mathrm{\ell }(x)-\mathrm{\ell }(y)\mathrm{\mid }\le ({\sum }_i\mathrm{\mid }{a}_i\mathrm{\mid })\parallel x-y{\parallel }_2$, so it is continuous. Its maximum $M>0$ is attained on the nonempty compact cell $C_p$ by [F8], and the set $G=\{x\in C_p:\mathrm{\ell }(x)=M\}$ is a face of $C_p$ by [F1], nonempty, bounded and closed, and of dimension strictly smaller than $\dim C_p$ because it lies in the proper affine subspace $\{\mathrm{\ell }=M\}\cap \mathrm{aff}{C}_p$. Iterating this construction strictly decreases the dimension, so it produces a $0$-dimensional face, which is a face of $C_p$ by [F1]. This gives the vertices, and their average $b_p$ is defined. If $b_p$ were not relatively interior, then by [F2] it would lie in a proper face $G$ of $C_p$, cut out by some active defining inequalities by [F1], and since $G$ is proper some active inequality $\mathrm{\ell }$ is not identically zero on $C_p$; then $\mathrm{\ell }\ge 0$ on $C_p$ and $0=\mathrm{\ell }(b_p)$ is an average of nonnegative numbers, so $\mathrm{\ell }$ vanishes at every vertex of $C_p$. But $\mathrm{\ell }$ is nonconstant on $C_p$, so its maximum face $\{\mathrm{\ell }={\max }_{C_p}\mathrm{\ell }\}$ is a face of $C_p$ on which $\mathrm{\ell }>0$, and by the previous paragraph that face contains a vertex of $C_p$, where $\mathrm{\ell }$ would have to vanish: a contradiction. Hence $b_p\in \mathrm{relint}{C}_p$.

Fix $p$. The faces of $C_p$, together with the empty set, form a finite convex cell complex: by [F1] each face is again a cell, faces of faces are faces, and the intersection of two faces is a face of both because it is obtained by activating the union of their defining equalities. Applying [F3] to this complex with the relative interior points $b_F$ of [step 1.1] gives a compatible finite simplicial triangulation of $C_p$. By induction on dimension over the coning construction its simplices are exactly the convex hulls $\mathrm{conv}(b_{F_0},\dots ,b_{F_k})$ for strict chains $F_0>F_1>\cdots >F_k$ of nonempty faces of $C_p$, and two such simplices with the same top face meet in the simplex of the common subchain. Moreover the construction only uses the faces and their barycentres, so for $r\le p$ the triangulations of $C_p$ and of $C_r$ agree on the common face $h_{r,p}(C_r)$, transported by the affine isometry $h_{r,p}$.

The family of all convex hulls $\mathrm{conv}(b_{F_0},\dots ,b_{F_k})$ over strict chains $F_0>\cdots >F_k$ in $P^{\ast }$ is a well-defined family of subsets of $X$, the map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:\mathrm{\mid }K\mathrm{\mid }\to X$ that is affine on each simplex of $K$ with the corresponding vertex values is well defined, and it is a bijection. Each such hull lies in the cell $C_{F_0}$ and is computed there; if the same chain is regarded inside a larger cell the hull is unchanged because all its points lie in the face $C_{F_0}$ and that face is convex, so the subset of $X$ is unambiguous. The images of the simplices of $K$ are exactly the simplices of the triangulations of the cells of [step 2.1], so they cover $X$, and distinct ones have disjoint relative interiors: within one cell this is the triangulation property of [step 2.1]; and if $x$ lies in the relative interior of simplices with top faces $F$ and $G$, then $x\in \mathrm{relint}F\cap \mathrm{relint}G$ by the coning description of [step 2.1], while $x\in C_F\cap C_G=C_{F\wedge G}$ by the intersection condition of [F6]; as $C_{F\wedge G}$ is a face of $C_F$ containing $x$ and $x$ is relatively interior in $F$, that face must be $F$ itself, and likewise $G\subseteq C_{F\wedge G}$, so $F$ and $G$ are faces of one cell with a common relative interior point and hence coincide by [F2]; then the two simplices are two cells of one triangulation with a common relative interior point and hence coincide. Therefore every point of $X$ is in the relative interior of exactly one of the simplices, so $\mathrm{\Phi }$ is bijective, and its carrier is unique.

There is a real $L$ with $1\le L<\mathrm{\infty }$, depending only on the finite model list, such that for every simplex of $K$ and every vertex $v$ the function ${\lambda }_v$ restricted to that simplex is $L$-Lipschitz for the Euclidean metric of its image. On a singleton simplex all coordinates are constant, with slope $0$. On a positive-dimensional simplex of $K$ with chain $F_0>\cdots >F_k$ the function ${\lambda }_v$ either vanishes identically, when $v$ is not one of the $F_i$, or equals the barycentric coordinate at $b_v$; its linear part has norm $1\mathrm{/}h$, where $h>0$ is the distance from $b_v$ to the affine hull of the remaining barycentres, because its gradient is perpendicular to that affine hull and the coordinate changes from $0$ there to $1$ over the perpendicular displacement of length $h$. Each such configuration is isometric to a configuration in a model cell, because the cells of $X$ fall into finitely many isometry classes and the face isometries are affine and isometric, so the positive-dimensional model simplices supply only finitely many positive numbers $h$. Take $L$ to be the maximum of $1$ and their reciprocals; when no such simplex exists take $L=1$. This bounds every coordinate slope.

The bijection $\mathrm{\Phi }$ of [step 3.1] is a homeomorphism from the weak topology of $\mathrm{\mid }K\mathrm{\mid }$ to the weak topology of $X$. A subset $V\subseteq X$ is weakly open in $X$ exactly when its trace on every closed cell is relatively open, and by [F4] applied to the finite triangulation of a cell this holds exactly when its trace on every simplex of the triangulation of every cell is relatively open. By [step 2.1] those simplices are the images under $\mathrm{\Phi }$ of the simplices of $K$, affinely and hence homeomorphically, so this is exactly the condition that ${\mathrm{\Phi }}^{-1}(V)$ has relatively open trace on every simplex of $K$, which by [F5] is openness in $\mathrm{\mid }K\mathrm{\mid }$. Hence $\mathrm{\Phi }$ and ${\mathrm{\Phi }}^{-1}$ carry open sets to open sets.

Every point of $X$ lies in the relative interior of exactly one simplex of $K$, its carrier, of dimension at most $D$. Uniqueness and existence are [step 3.1]. A simplex of $K$ is a strict chain $F_0>\cdots >F_k$ of nonempty faces; passing from $F_{i+1}$ to $F_i$ is passing to a proper face, which by [F2] lies in a supporting hyperplane, so the dimensions strictly increase along the chain: $k\le \dim F_0\le D$; the simplex therefore has $k+1\le D+1$ vertices and dimension at most $D$.

Define ${\lambda }_v(x):=\alpha (v)$ where $x=\mathrm{\Phi }(\alpha )$, using the bijection of [step 3.1]. Each ${\lambda }_v$ is well defined, takes values in $[0,1]$ by [F5], is affine on every simplex of $K$ because $\alpha \mapsto \alpha (v)$ is affine on each simplex and $\mathrm{\Phi }$ is affine there, and satisfies ${\sum }_v{\lambda }_v(x)=1$: the sum is over the support of the carrier of $x$, a finite set, with total $1$ by [F5].

Let $x,y$ lie in a common cell $C_p$, whose Euclidean metric is $d_p$. Then $\mathrm{\mid }{\lambda }_v(x)-{\lambda }_v(y)\mathrm{\mid }\le L{d}_p(x,y)$ for every $v$. The segment $[x,y]$ lies in $C_p$ by convexity, and it is covered by the finitely many simplices of the triangulation of $C_p$; its intersection with a simplex is convex, hence a point or a subsegment, and on each nondegenerate subsegment ${\lambda }_v$ is $L$-Lipschitz by [step 3.2]. Summing over the finitely many subsegments gives $\mathrm{\mid }{\lambda }_v(x)-{\lambda }_v(y)\mathrm{\mid }\le L{d}_p(x,y)$.

For all $x,y\in X$ and every vertex $v$ one has $\mathrm{\mid }{\lambda }_v(x)-{\lambda }_v(y)\mathrm{\mid }\le Ld(x,y)$. Let $x=x_0,\dots ,x_m=y$ be a chain as in [F6]; each consecutive pair lies in a common cell, so [step 4.4] gives $\mathrm{\mid }{\lambda }_v(x_{i-1})-{\lambda }_v(x_i)\mathrm{\mid }\le L{d}_{p_i}(x_{i-1},x_i)$, and summing the at most $m$ inequalities and using the triangle inequality for real numbers gives $\mathrm{\mid }{\lambda }_v(x)-{\lambda }_v(y)\mathrm{\mid }\le L\mathrm{\ell }(x_0,\dots ,x_m)$. Taking the infimum over all chains from $x$ to $y$ gives the claim, since $L>0$.

Let $x\in X$ with carrier simplex $\sigma$ and let $\delta :=1\mathrm{/}(2L(D+1))>0$, which is legitimate because $L\ge 1$. By [step 4.2] the carrier has at most $D+1$ vertices and the coordinates $({\lambda }_v(x){)}_v$ of the carrier are nonnegative and sum to $1$ over them, so some vertex $v$ of $\sigma$ satisfies ${\lambda }_v(x)\ge 1\mathrm{/}(D+1)$. For every $y\in X$ with $d(x,y)<\delta$ we get ${\lambda }_v(y)\ge {\lambda }_v(x)-Ld(x,y)>1\mathrm{/}(D+1)-1\mathrm{/}(2(D+1))>0$ by [step 5.1]; so the support of the carrier of $y$ contains $v$, which means that $y$ lies in the relative interior of a simplex containing $v$, that is, in the open star of $v$ by [F7]. Hence $B(x,\delta )$ is contained in the open star of $v$.

For every vertex $v$ the closed star of $v$ is compact and metrizable, and its open star is open and hence a neighbourhood of each of its own points. The closed star is $\mathrm{\Phi }(\mathrm{\mid }S\mathrm{\mid })$ where $S$ is the subcomplex of $K$ consisting of every simplex containing $v$ together with all its faces: the simplices containing $v$ are chains in $P^{\ast }$ containing $v$, and the elements of $P^{\ast }$ comparable with $v$ are finitely many, because $P_{\le v}$ is finite by the shape condition of [F6] and the faces $q\ge v$ are finitely many by (H2) of [F6] (a cell $C_q$ meets the relative interior of $C_v$ exactly when $v\le q$). There are finitely many such chains, and each has finitely many faces, so $S$ is a finite subcomplex, $\mathrm{\mid }S\mathrm{\mid }$ is compact and metrizable by [F4], and its image under the homeomorphism of [step 4.1] is compact and metrizable. The open star is the union of the relative interiors of precisely the simplices containing $v$, not of all faces in $S$. Equivalently it is $\{x:{\lambda }_v(x)>0\}$, which is open: on each cell the coordinate is continuous by step 4.4, or for the chain metric it is Lipschitz by step 5.1. It is contained in the closed star and is a neighbourhood of each of its own points; and every $x\in X$ lies in the relative interior of its carrier, which has at least one vertex, so the open stars cover $X$. Finally, if a vertex $w$ lies in a common cell with $v$, then $w\le q$ for some $q\ge v$; there are finitely many such $q$ as just shown and each $P_{\le q}$ is finite, so only finitely many such $w$ exist.