Relative smooth triangulation and bundle fitting

Uniform rank bound. If $K$ has only vertices, immersion is automatic and a sufficiently small value error preserves the finite separation of distinct embedded vertices; the empty case is vacuous. Otherwise, for each positive-dimensional original simplex $\sigma$ let $D_{\sigma }$ be its direction space. Its tangent cone at $b\in \sigma$ spans $D_{\sigma }$ and every vector of $D_{\sigma }$ is a difference of two cone vectors, so whole-star injectivity makes $A_{b,\sigma }:=D(f\mathrm{\mid }\sigma )(b)$ injective on $D_{\sigma }$. Compactness of $\sigma$ times its unit sphere gives a common least singular value $\mu >0$ over all original simplices. The strong derivative bound also holds, with $\le \delta$, on the entire direction space of each refined simplex: apply the bound at its relative interior points, where every direction is tangent, then pass to its boundary by continuity. Thus $D(g\mathrm{\mid }s)(b)$ has least singular value at least $\mu -\delta$ on every refined simplex $s$, and derivative errors less than $\mu \mathrm{/}2$ give angle errors at most $4\delta \mathrm{/}\mu$ by [F4].

Two-segment separation. Suppose two segments have common endpoint $x$ and a simplexwise affine injective map $H$ carries their outward directions to nonzero vectors making angle $\beta >0$. In their image plane project onto the oriented line perpendicular to the internal bisector, or onto the image line when $\beta =\pi$. The two projected directions have opposite signs. Any nonzero derivative making angle less than $\beta \mathrm{/}2$ with its corresponding image direction has the same strict projection sign. Segment integration then makes the projected map strictly monotone along each segment and puts their open images on opposite sides of the image of $x$, so the map is injective on their union. This also applies to a finite piecewise smooth subdivision of the segments: integrate on each subsegment and add.

Geometric separation. For two geometric simplices ${\sigma }_1,{\sigma }_2$ with common face $\tau$ proper in both, let $c$ be the barycentre of $\tau$ and $\eta$ the opposite face of ${\sigma }_1$. For $p=(1-t)x+ty$, $x\in \tau$, $y\in \eta$, set $X(p)=(1-t)x+tc$, taking $X(p)=c$ when $t=1$ and $X(p)=p$ on $\tau$. This is a continuous projection fixing $\tau$, and $p-X(p)=t(y-c)$. The unit directions of ${\sigma }_2-c$ form a compact set: in barycentric direction coordinates the cone is given by zero sum and nonnegative coordinates at vertices outside $\tau$, hence is closed. No ray $y-c$ with $y\in \eta$ has a direction in this cone, since a common ray would either put $y$ in $\tau$ or put $y$ in $\mathrm{aff}(\tau )$, contrary to [F5]. Therefore the angle between those directions has a positive minimum. For $q\in {\sigma }_2$ the vector $q-X(p)$ belongs to that cone, because $c-X(p)$ lies in the direction space of $\tau$, the lineality space of the cone. This proves the same lower angle bound for the two rays from $X(p)$ to $p$ and $q$, when both are nonzero. The projection formula is preserved by separate affine injections of the two simplices agreeing on $\tau$, provided their image intersection is exactly the image of $\tau$.

Uniform frozen angles. For $b\in \tau$ define the frozen map on each simplex containing $b$ by $H_b(x)=A_{b,\sigma }(x-b)$. This is simplexwise affine, agrees on common faces, and is injective on their union by the whole-star hypothesis. Its image simplices therefore intersect exactly in $H_b(\tau )$. Apply step 1.3 to these image simplices. Their separation angle has a positive minimum uniformly over $b\in \tau$: use the fixed compact sets $y\in \eta$ and unit directions $z$ of ${\sigma }_2-c$, and minimize the continuous angle between $A_{b,{\sigma }_1}(y-c)$ and $A_{b,{\sigma }_2}z$ on their product with $\tau$. Neither vector vanishes by step 1.1, and a zero angle is excluded by the image intersection property. The finite minimum over all ordered original simplex pairs, capped at $\pi \mathrm{/}2$, gives $\alpha >0$; take $\alpha =\pi \mathrm{/}2$ if there are no such pairs. No single linear extension of $H_b$ to the span of two simplices is asserted.

Local estimates independent of the subdivision. Choose $\delta <\mu \mathrm{/}2$ with $4\delta \mathrm{/}\mu <\alpha \mathrm{/}8$. For each $b$, choose a small relative ball $U_b$ meeting only original simplices containing $b$. Continuity of their finitely many derivative fields makes the angle between $D(f\mathrm{\mid }\sigma )(c)h$ and $A_{b,\sigma }h$ less than $\alpha \mathrm{/}8$ for $c\in U_b\cap \sigma$ and every nonzero direction $h\in D_{\sigma }$. Step 1.1 and [F4] give angle less than $\alpha \mathrm{/}4$ between each refined derivative of $g$ along $h$ and $A_{b,\sigma }h$. This bound holds on subdivided segments, including their one-sided derivatives. Shrink to a ball $V_b$ so that for any two original simplices containing $b$, the projection in step 1.3 sends $V_b\cap {\sigma }_1$ into $U_b$; there are only finitely many projections and each fixes $b$. The two segments joining points of $V_b$ through that projection lie in $U_b$, since its intersections with simplices are convex. These neighbourhoods depend on $f$ and $K$, not on $g$ or its subdivision.

Local injectivity and immersion. For distinct $p,q\in V_b$ in one original simplex, join them through their midpoint; their frozen outward rays are antiparallel. Otherwise use their minimal original carriers and the projection of step 1.3 onto their common face; both carriers contain $b$, and their frozen rays make angle at least $\alpha$ by step 2.1. If one segment is trivial, the other alone has a strictly monotone projection. In all cases steps 1.2 and 3.1 give $g(p)\mathrm{\ne }g(q)$. To check immersion at $b$, apply the same argument to the frozen piecewise linear map $J_b(x)=d{g}_b(x-b)$ on the refined tangent fan. Scale any two star vectors until their endpoints lie in $V_b$ and in the closed star of $b$ in the subdivision. On every refined cone the derivative of $J_b$ is $D(g\mathrm{\mid }s)(b)$, whose angle from the appropriate $A_{b,\sigma }$ is less than $\alpha \mathrm{/}8$ by step 1.1; the projected paths stay in the original cones containing $b$ and may be split across the finitely many refined cones. Step 1.2 separates their endpoints exactly as above. Hence $d{g}_b$ is injective on the entire tangent star, for every $b$, proving immersion.

Global embedding. If $f$ is an embedding, take a finite subcover $V_{b_1},\dots ,V_{b_m}$ of $\mathrm{\mid }K\mathrm{\mid }$ and put $C=\mathrm{\mid }K{\mathrm{\mid }}^2\setminus {\bigcup }_i(V_{b_i}\times V_{b_i})$. This compact set misses the diagonal, so $\mathrm{\mid }f(p)-f(q)\mathrm{\mid }$ has a positive minimum $\eta$ on it; if it is empty omit this bound. Require also $\delta <\eta \mathrm{/}3$. On $C$, $\mathrm{\mid }g(p)-g(q)\mathrm{\mid }>\eta -2\delta >0$, and outside $C$ distinct points are separated by step 4.1. Thus $g$ is continuous and injective on a compact space, hence a homeomorphism onto its image by [F2]; together with immersion this is an embedding. The same $\delta$ works for every finite subdivision, with no shape or mesh restriction.

Lattice subdivisions with a uniform thickness bound. If $K$ has no positive-dimensional simplices, its unchanged complex and secant $f$ prove the approximation assertions with the thickness condition vacuous. Otherwise give the finitely many vertices a total order and realize the abstract complex as a union of faces of one standard simplex $\mathrm{\Delta }\subset {\mathbb{R}}^N$ with vertices $0,e_1,\dots ,e_N$. The original geometric realization is related to this one by a linear isomorphism on each simplex, and finitely many such isomorphisms have uniformly bounded operator norms and inverse norms on their affine spans. In ${\mathbb{R}}^N$ cut the unit lattice cubes by the hyperplanes ${\sum }_i{x}_i=j$, $j\in \mathbb{Z}$, and all their faces; these cuts form a locally finite convex cell complex, since intersections arise by activating the shared supporting equalities. After translating a cube to $[0,1{]}^N$ only the cuts with $j=0,\dots ,N$ occur, so there are finitely many cell shapes up to translation; triangulating every cell compatibly by [F1] with the translation-invariant choice of the average of its vertices gives finitely many simplex shapes, with maximal diameter $D$ and positive minimum thickness ${\theta }_0$ in positive dimensions. The simplex $m\mathrm{\Delta }=\{x_i\ge 0,{\sum }_i{x}_i\le m\}$ and each of its faces are unions of these cells, so scaling by $1\mathrm{/}m$ gives subdivisions of $\mathrm{\Delta }$, hence of the embedded abstract complex, with mesh at most $D\mathrm{/}m$ and thickness at least ${\theta }_0$; restrictions to all faces agree because one global cell complex is used. Transporting back to the original realization multiplies diameter and inradius by the finitely many bounded factors of the linear isomorphisms, leaving the claimed mesh bound and one positive thickness $\theta$. Iterated barycentric subdivision has no such bound and is not used.

Combinatorial extension of a subdivision, and coned corrections. For the extension statement, extend the given subdivision of $A$ combinatorially in increasing dimension: keep every original simplex outside the closed star of $A$, and for each affected simplex whose boundary has already been subdivided, cone that boundary subdivision from the original barycentre. Each cone over a face simplex from the barycentre of the ambient simplex is a simplex, and the cones agree on common faces by the ray description of a simplex, so this produces a subdivision of $K$ containing the given subdivision of $A$. Suppose a correction $e=g-f$ is already defined and piecewise smooth on the subdivided boundary of an original simplex $\sigma$ with barycentre $c$; every $x\mathrm{\ne }c$ in $\sigma$ has the unique expression $x=c+t(y-c)$ with $0<t\le 1$ and $y\in \mathrm{\partial }\sigma$. The function $b(s)=\exp (-1\mathrm{/}s)$ for $s>0$ and $b(s)=0$ for $s\le 0$ is smooth, since every right derivative is a polynomial in $1\mathrm{/}s$ times $\exp (-1\mathrm{/}s)$ and tends to zero at $0$. Thus $a(t)=b(t-1\mathrm{/}3)\mathrm{/}(b(t-1\mathrm{/}3)+b(2\mathrm{/}3-t))$ is a smooth scalar taper with $a(t)=0$ for $t\le 1\mathrm{/}3$ and $a(t)=1$ for $t\ge 2\mathrm{/}3$ and define $G(x)=f(x)+a(t)e(y)$ on $\sigma \setminus \{c\}$ and $G(c)=f(c)$. On each coned simplex this is smooth across $c$, because $a=0$ near $c$ forces $G=f$ there, and it assumes the prescribed values on the boundary; faces agree because $t$ and $y$ are determined by $x$ and $e$ is one function on the subdivided boundary.

Secant estimate on a shape-controlled simplex. A zero-dimensional refined simplex has secant value $f$ and zero tangent space, so contributes zero error. Let $s$ be a positive-dimensional refined simplex inside an original simplex $\sigma$, with incenter $b$, diameter $h$ and inradius $r\ge \theta h$, and put $T_b(x)=f(b)+Df(b)(x-b)$. On the compact simplex $\sigma$ the derivative of $f$ has a modulus of continuity ${\omega }_{\sigma }(h)\to 0$ as $h\to 0$, uniformly in the base point. To justify uniformity, for any $\epsilon >0$ continuity supplies balls about each base point on which its derivative differs from the centre value by less than $\epsilon \mathrm{/}2$. The balls of half those radii cover $\sigma$; a finite subcover and the minimum of their half-radii ensure that any two sufficiently close points have derivative difference less than $\epsilon$ by the triangle inequality. Define ${\omega }_{\sigma }(h)$ as the supremum of those differences at distance at most $h$; boundedness follows from the same finite-cover argument. Segment integration along $[b,x]$ gives $\mathrm{\mid }f(x)-T_b(x)\mathrm{\mid }\le h{\omega }_{\sigma }(h)$ for $x\in s$. The affine map $S_m-T_b$ interpolates the vertex errors $f(v)-T_b(v)$, each of modulus at most $h{\omega }_{\sigma }(h)$, and an affine function on a simplex attains its supremum at a vertex, so ${\sup }_s\mathrm{\mid }{S}_m-T_b\mathrm{\mid }\le h{\omega }_{\sigma }(h)$. For an affine map $A$ on $s$ with ${\sup }_s\mathrm{\mid }A\mathrm{\mid }\le E$ and a unit vector $v$ in the affine span of $s$, the point $b+rv$ lies in $s$ by the definition of incenter and inradius, so $\mathrm{\mid }DA(v)\mathrm{\mid }=\mathrm{\mid }A(b+rv)-A(b)\mathrm{\mid }\mathrm{/}r\le 2E\mathrm{/}r$. Applying this to $S_m-T_b$ gives $\mathrm{\parallel }D{S}_m-Df(b)\mathrm{\parallel }\le 2{\omega }_{\sigma }(h)\mathrm{/}\theta$, and adding the continuity term $\mathrm{\parallel }Df(b)-Df(x)\mathrm{\parallel }\le {\omega }_{\sigma }(h)$ gives $\mathrm{\parallel }D{S}_m-Df(x)\mathrm{\parallel }\le (2\mathrm{/}\theta +1){\omega }_{\sigma }(h)$ for all $x\in s$. As the mesh tends to zero, both the value and the derivative errors tend to zero uniformly over all base points and all vectors of the refined tangent cones of $\sigma$; taking the maximum of the finitely many moduli ${\omega }_{\sigma }$ over the original simplices proves strong piecewise $C^1$ convergence.

Extension error bounds. On a cone over a boundary simplex, $x=c+t(y-c)$; writing the barycentric coordinates of $x$ in $\sigma$ and summing those outside the face carrying $y$ shows that $t$ is an affine function of $x$ divided by the fixed barycentre-to-face height, and $y=c+(x-c)\mathrm{/}t$. On the region $t\ge 1\mathrm{/}3$ the bounds $\mathrm{\mid }Dt\mathrm{\mid }\le C_1(\sigma )$ and $\mathrm{\parallel }Dy\mathrm{\parallel }\le C_2(\sigma )$ depend only on the original simplex, not on the subdivision of the boundary. The strong bound on $De$ controls its operator norm on the entire direction space of each refined boundary simplex: apply it at relative interior points, where every direction is tangent, and pass to the boundary by continuity. The correction of step 1.2 has value $a(t)e(y)$ and derivative $a^{\mathrm{\prime }}(t)e(y)Dt+a(t)De(y)Dy$ by the chain rule, so its supremum norm and derivative norm are bounded by $C_{\sigma }\mathrm{\parallel }e{\mathrm{\parallel }}_{C^1}$ with $C_{\sigma }$ depending only on $\sigma$; near $c$ the correction vanishes identically. Hence, for finitely many affected simplices handled in increasing dimension, the product of the finitely many constants $C_{\sigma }$ converts any requested final error into a sufficient bound on the initial error, uniformly over all subdivisions of $A$; a zero correction on an already fixed subcomplex is retained, and the correction is zero outside the closed star of $A$ by construction. If $p$ is affine and its linear part is $P$, then $p(g)=p(f)$ is equivalent to $P(e)=0$ on $A$. The formula gives $P(a(t)e(y))=a(t)P(e(y))=0$ on every newly extended simplex, so induction preserves $p(G)=p(f)$ everywhere.

Relative approximation with plane preservation. For the relative statement, barycentrically subdivide once if necessary so that $L$ is full: a face chain all of whose barycentric vertices lie in $L$ lies in $L$ by [F1]. Then let $d$ be the sum of the barycentric coordinates at vertices outside $L$; on each subdivision simplex the coordinates are affine, so $d$ is a global continuous piecewise affine nonnegative function, and its zero set is exactly $\mathrm{\mid }L\mathrm{\mid }$: a point of $\mathrm{\mid }L\mathrm{\mid }$ lies in a subdivision simplex all of whose vertices lie in $L$ because $L$ is full, so all coordinates outside $L$ vanish, while $d(x)=0$ forces $x$ into the hull of vertices lying in $L$, hence into $\mathrm{\mid }L\mathrm{\mid }$. The zero set of $d$ is closed and the complement of $U$ is compact, so, when $\mathrm{\mid }K\mathrm{\mid }\setminus U$ is nonempty, a positive lower bound $d_0$ exists: continuity and positivity give neighbourhoods on which $d$ is at least half its value at the centre, and a finite subcover gives the minimum of finitely many positive half-values. If $\mathrm{\mid }K\mathrm{\mid }\setminus U$ is empty, take $g_m=f$ and the assertion outside $U$ is vacuous. Otherwise take the cutoff $\chi (s)=a(s\mathrm{/}{d}_0)$ using step 1.2, so $\chi =0$ on $[0,d_0\mathrm{/}3]$ and $\chi =1$ on $[2d_0\mathrm{/}3,\mathrm{\infty })$ and its values lie in $[0,1]$; its derivative is bounded, and the piecewise affine function $d$ has bounded derivative on the fixed finite subdivision. Setting $g_m=f+\chi (d)(S_m-f)$ gives, near $\mathrm{\mid }L\mathrm{\mid }$, $g_m=f$; outside $U$, $g_m=S_m$; and the derivative error is bounded by the derivative error of $S_m$ plus a fixed constant times its value error, both of which tend to zero by step 2.1. Each simplex restriction is smooth and face values agree. If $f$ maps an original simplex into an affine plane, then so do its vertex interpolants and every affine combination of $f$ with them, so all affine target-plane constraints are preserved, and the preliminary subdivision still admits the uniformly controlled subdivisions of step 1.1.

Assembly. The subdivisions of step 1.1 have mesh tending to zero and one positive thickness bound and, by step 2.1, their secant maps converge strongly in piecewise $C^1$ to $f$; step 3.1 provides the relative version, fixing $f$ near $L$, agreeing with a secant map outside $U$ and preserving affine target-plane constraints. For the extension statement, step 1.2 extends the subdivision and defines the coned correction, and step 2.2 bounds its $C^1$ norm by a constant depending only on the original simplices times the $C^1$ norm of the boundary correction, uniformly over all subdivisions of $A$; choosing the initial error small enough, in the finitely many affected simplices in increasing dimension, therefore realises any prescribed final strong-approximation error, keeps the correction zero outside the closed star of $A$, and preserves a vanishing affine-linear component. This proves all three assertions.

Structure of a zero set. Fix a coordinate and an original simplex, writing $u={\rho }_{i}F$ and $Z=u^{-1}(0)\cap \sigma$. The zero set is a finite union of faces. Its maximal faces are pairwise disjoint: if two met at $b$, the hypothesis would identify their union of tangent cones with the zero face of the convex cone $C_b$ under the supporting form $d{u}_b$. This zero face is convex and corresponds to a face $\tau$ of $\sigma$ containing $b$: the supporting form vanishes on the direction space of the minimal face containing $b$, is nonnegative on every vertex direction $v-b$, and its zero cone is generated by that direction space and precisely the zero-valued vertex directions. A relative interior direction of that face must belong to one of the finitely many zero-face cones, forcing that face to contain $\tau$; hence $\tau \subset Z$, contrary to maximality of two distinct intersecting faces. Thus, near any maximal zero face $\tau$, the only zero face is $\tau$ itself. If $Z=\sigma$, plane preservation already forces ${\rho }_{i}G=0$ throughout $\sigma$; assume otherwise that $Z$ is nonempty and proper. For every $z\in \tau$ and every vertex $v$ outside $\tau$, the tangent hypothesis gives $d{u}_z(v-z)>0$. Their finite minimum over the compact face $\tau$ is a number ${\kappa }_{\tau }>0$. This is a bound on those transverse vertex directions, not on all unit directions outside the zero cone.

Distance lower bound. Near a proper maximal zero face $\tau$, write $x=(1-t)z+ty$ by barycentric projection, with $z\in \tau$, $y$ in the opposite face and $t$ the sum of coordinates outside $\tau$. For $x$ sufficiently near $\tau$, $t<1\mathrm{/}2$ and $\mathrm{\mid }x-z\mathrm{\mid }\le t\mathrm{diam}\sigma$ is arbitrarily small, uniformly along $\tau$. Step 1.1 gives $d{u}_z(y-z)\ge {\kappa }_{\tau }$ by expressing $y$ as a convex combination of outside vertices. Uniform continuity of $du$ on $\sigma$ follows directly by covering it with half-radius continuity balls, taking a finite subcover, and using the triangle inequality on each corresponding full-radius ball. It gives $d{u}_{z+s(x-z)}(y-z)\ge {\kappa }_{\tau }\mathrm{/}2$ for $0\le s\le 1$, after choosing the neighbourhood small enough. Segment integration yields $u(x)\ge t{\kappa }_{\tau }\mathrm{/}2\ge [{\kappa }_{\tau }\mathrm{/}(2\mathrm{diam}\sigma )]\mathrm{dist}(x,Z)$, since $z\in Z$ and $\mathrm{\mid }x-z\mathrm{\mid }\le t\mathrm{diam}\sigma$. Take the finite minimum of these positive constants over the maximal zero faces. If the compact remainder outside their neighbourhoods is nonempty, the continuous positive ratio $u(x)\mathrm{/}\mathrm{dist}(x,Z)$ has a positive minimum there by [F2]; if it is empty, omit this bound. Consequently there is $a>0$ with $u(x)\ge a\mathrm{dist}(x,Z)$ on the whole simplex. No such bound is needed when $Z$ is empty or the whole simplex.

Perturbing the coordinate functions. For $x\in \sigma$ choose a closest point $z\in Z_{i,\sigma }$ when the zero set is nonempty and proper; it exists by compactness of $Z_{i,\sigma }$ and the segment $[z,x]$ lies in the convex simplex. Since ${\rho }_{i}G(z)={\rho }_{i}F(z)=0$ on the zero subcomplex, segment integration of the piecewise smooth difference along $[z,x]$, split into the finitely many refined pieces and added, gives $\mathrm{\mid }{\rho }_{i}G(x)-{\rho }_{i}F(x)\mathrm{\mid }\le \delta \mathrm{\mid }x-z\mathrm{\mid }=\delta \mathrm{dist}(x,Z_{i,\sigma })$, using the derivative bound of the strong approximation. Choose $\delta <a\mathrm{/}2$ with $a$ from step 2.1. Then ${\rho }_{i}G(x)\ge (a\mathrm{/}2)\mathrm{dist}(x,Z_{i,\sigma })$, so ${\rho }_{i}G\ge 0$ on $\sigma$ with equality exactly on $Z_{i,\sigma }$. If $Z_{i,\sigma }=\sigma$, plane preservation gives ${\rho }_{i}G=0$ there. If $Z_{i,\sigma }$ is empty, then ${\rho }_{i}F$ has a positive minimum $m_{\sigma }$ on the compact simplex by [F2], and the value error $\mathrm{\mid }G-F\mathrm{\mid }<\delta$ with $\delta <m_{\sigma }\mathrm{/}2$ keeps ${\rho }_{i}G$ positive on all of $\sigma$.

Conclusion. Choose $\delta >0$ below ${\delta }_0$ of [F1], every $a\mathrm{/}2$ for a proper nonempty zero set, and every $m_{\sigma }\mathrm{/}2$ for an empty zero set, omitting nonexistent cases. Every strong $\delta$-approximation $G$ preserving the zero subcomplexes then has ${\rho }_{i}G\ge 0$ with zero set exactly $Z_{i,\sigma }$ on every simplex, for every $i$: all coordinate functions are nonnegative and the coordinate zero sets are exactly the prescribed ones, so $G$ maps $K$ into the same closed orthant with the same boundary structure. By [F1] $G$ is again a whole-star embedding. This supplies the side control that affine-plane preservation alone does not give, as the half-plane warning in the cited Munkres exercise requires.

Compactness and margin facts. A finite geometric realization is compact by [F3]: the realization is a continuous image of the finite disjoint union of its compact closed simplices, and inverse images of closed conditions are finite unions of closed simplex traces. Continuous images of compact sets are compact, because an open cover pulls back to an open cover with a finite subcover, and a closed subset of a compact set is compact by adjoining its open complement to a cover. In a local Euclidean chart a compact set is closed, so for a nonempty compact set $A$ disjoint from a nonempty closed set $C$ a positive gap follows directly from compactness: the pairs $(a,r)$ with $B(a,3r)$ disjoint from $C$ cover $A$ by balls $B(a,r)$, a finite subcover $B(a_j,r_j)$ exists, and the triangle inequality then gives $d(x,y)>2r_j$ for $x\in A$, $y\in C$ and a covering $j$, so $2{\min }_j{r}_j>0$ is a gap; empty sets impose no constraint. For a compact source with an open chart cover, [F3] gives a positive Lebesgue margin: choosing a finite cover by balls $B(a_j,r_j)$ with $B(a_j,3r_j)$ inside chart members, any set of diameter less than ${\min }_j{r}_j$ that meets $B(a_j,r_j)$ lies in $B(a_j,2r_j)$ and hence in the same chart. By [F3] a closed vertex star has diameter at most twice the mesh and $k$ successive star neighbourhoods at most $2(k+1)$ times the mesh, so a sufficiently fine subdivision puts all finitely many required support stars inside their assigned chart preimages and keeps the protected-region separation; all of this uses only finitely many explicitly chosen balls.

Euclidean fitting step. Two finite linear embedded complexes in ${\mathbb{R}}^q$ can be placed in one finite convex cell complex even when their underlying sets differ: cut a bounding box by all affine hyperplanes defining the affine hulls and barycentric facet inequalities of the finitely many simplices of both complexes. Each resulting cell is determined by choosing one weak side of each hyperplane; intersections activate further equalities and are common faces; each original simplex is a union of cells because its equalities and weak inequalities are among the cuts, and closed cells outside a simplex can meet it only along common faces. Triangulate this complex compatibly by [F4] and restrict to the two original polyhedra: their intersection is now a subcomplex, and one barycentric subdivision makes it full, because a face chain all of whose barycentric vertices lie in the intersection has its largest face there and hence the whole simplex there. Pulling these subdivisions back through the linear embeddings gives fitting in a full subcomplex by arbitrary-overlap refinement; no same-polyhedron common-refinement theorem is used beyond its scope.

Local fitting of an attached piece. Work in ${\mathbb{R}}^q$ with a finite complex $P=Q\cup A$ and a piecewise smooth map $T$ such that $T{\mathrm{\mid }}_Q$ and $T{\mathrm{\mid }}_A$ are embeddings, $A$ being the new piece. Let $P_0$ contain the closed stars of all simplices meeting $T^{-1}(T(A))$ and let $P_1$ be the closed star of $P_0$; then $T^{-1}(T(A))$ lies in the relative interior of $\mathrm{\mid }{P}_0\mathrm{\mid }$, so by step 1.1 the compact image of the complement of that relative interior has positive distance from the compact set $T(A)$, unless the complement is empty, in which case the distance constraint is omitted. Choose the approximation tolerance below a quarter of that distance and small enough for [F1] on the two embeddings separately, and first use the shape-controlled secant assertion of [F2] on the finite subcomplex $P_1$. Then use its local extension assertion to extend that small change over the closed star of $P_1$, leaving $T$ unchanged outside that star. The secant error on $P_1$ is chosen below the extension threshold for the required final error. Every new intersection of $T(Q)$ with $T(A)$ lies in $\mathrm{\mid }{P}_0\mathrm{\mid }$ by the distance bound and the triangle inequality, and both restricted maps remain embeddings. On $P_1$ the two maps are linear on a finite subdivision, so step 1.2 fits their images there. Pull the subdivisions back and extend them over the whole complexes by coning subdivided faces from original simplex barycentres, without changing the altered map. The resulting union is a whole-star embedding: at a point of the intersection a neighbourhood is an actual geometric linear complex whose differential is the translated affine realization, hence injective on the whole tangent star; at other points one is locally inside a single embedded piece, which is an embedding by [F1]; and the glued map is continuous and bijective from a compact space onto its image, hence a homeomorphism.

Affine-base bundle projection. In a product chart write $T(x)=(h(r(x)),v(x))$ with projection $p_1$ onto the base, and suppose first that $h$ is affine on every base simplex reached by the affected domain simplices. A secant of $T$ then still has first coordinate $hr$, because both $r$ and $h$ are affine there and the secant matches the vertex values; every boundary correction of [F2] has zero first coordinate, and the explicit correction formula multiplies that correction by a scalar, so it retains a zero first-coordinate component. Hence every perturbation of step 2.1 preserves $p_{1}T=hr$ exactly. At points identified by the fit, equality of the altered maps and injectivity of $h$ give equality of the $r$-values, so $r$ descends to the glued union and is affine on its simplices.

General base: shift and shift-back. Remove the affine hypothesis on $h$ by the shift construction. On a finite base subcomplex covering the affected projections and their stars, choose a secant approximation $h^{\mathrm{\prime }}$ to $h$ sufficiently close for [F1]; extend it to the required base neighbourhood by [F2]. Subdivide the domain so that $r$ maps each new simplex into one simplex of the secant subdivision of the base: intersect each domain simplex with inverse images of those base simplices, a finite convex cell decomposition, then triangulate it by [F4]. This necessary subdivision keeps the original point map $r$ affine. Replace $T$ temporarily by $T_1=(h^{\mathrm{\prime }}r,v)$. This is an embedding on each original embedded piece: equality of two $T_1$-values gives equal $r$-values by injectivity of $h^{\mathrm{\prime }}$, then equal $v$-values, hence equal original $T$-values. At the differential level the first component is $d{h}_{r(x)}^{\mathrm{\prime }}d{r}_x$, and equality on two tangent-star vectors forces equality of their $d{r}_x$-values by whole-star injectivity of $h^{\mathrm{\prime }}$; equality of the fibre components then gives equality under $d{T}_x$, hence equality of the original star vectors. Compactness gives the homeomorphism property, so $T_1$ is again a whole-star embedding. Apply step 3.1 to fit $T_1$, obtaining $T_2=(h^{\mathrm{\prime }}r,v^{\mathrm{\prime }})$, and shift back to $T_3=(hr,v^{\mathrm{\prime }})$. The same injectivity and differential argument shows that each embedding and the embedding of the fitted union survive the shift back, and two points have equal images under $T_3$ exactly when they have equal images under $T_2$, because $h$ and $h^{\mathrm{\prime }}$ are injective and the fibre coordinate is unchanged; hence the full-subcomplex identifications remain identical. The first-coordinate changes cancel completely, leaving the original diagram $\pi T_3=hr$. Choosing the temporary secant and the fitting errors successively small gives the prescribed final strong error, bounded through the chain rule on finitely many compact charts.

Finite induction over the new simplices. In a general smooth target, subdivide the new complex so that each closed simplex and the old pieces in its intersection neighbourhood lie in one coordinate chart; in the bundle case choose product charts and base charts with the same property, using the margins of step 1.1. Order the finitely many new simplices with faces first. Begin with the old embedding and suppose the already processed new simplices fit it in a full subcomplex and their union is an embedding; attach the next simplex along its already processed boundary, giving $P=Q\cup A$, and apply step 2.1, or the projection-preserving steps 3.1-4.1 in product charts. The changed map on the processed new subcomplex extends over the unprocessed simplices with arbitrarily small prescribed error by [F2], and in product charts its correction has zero base component, so the diagram is preserved. Choose the error at each of the finitely many stages small enough to retain the previous embeddings and fit margins, with the total error below the requested tolerance. Since only finitely many simplices and stages occur, no unevidenced limiting map or infinite subdivision arises.

Supports, protected regions and corner orthants. Every local change and its extension is supported on the closed stars used in that construction; if those supports have images in the prescribed open neighbourhood $V$ of the new piece and $\bar{V}\cap R=\mathrm{\varnothing }$ for the closed protected region $R$, then the parametrizations and diagrams on $R$ are untouched and pieces already fitted outside the supports keep their identifications, so the induction adds only the new fitted regions. For coordinate affine-plane constraints, secants remain in the same plane and the extension formula remains there because the correction lies in the translation space of the plane, and in the bundle shift the fibre coordinate is unchanged by either shift, so fibre-plane constraints persist. For a corner target, after each plane-preserving local change the image still lies in the chart orthant with the same coordinate zero sets by [F6], before the chart inverse is applied. This proves all the fitting, projection-preserving, supported and relative assertions under the stated separation and product-chart hypotheses.

Box pieces. If $M$ is empty take the empty complex; if $M$ has dimension zero, compactness makes it a finite discrete set whose points are the required vertices. Otherwise choose finitely many corner charts and closed coordinate boxes $Q_i=[0,R_i{]}^{c_i}\times [-R_i,R_i{]}^{n-c_i}$ contained in them whose smaller relative interiors cover $M$; at a point of smaller depth shrink the chart so that all remaining corner coordinates are strictly positive and are treated as ordinary coordinates. Each box has a finite cell triangulation by [F2], whose inverse chart map $F_i:K_i\to M$ is a whole-star embedding, being the restriction of an invertible smooth coordinate map to an actual geometric complex; its preimages of local corner planes are subcomplexes. In any other corner chart on an overlap, the transition and its inverse map the corresponding tangent orthant and its faces isomorphically onto the target orthant and faces, because their derivatives preserve the tangent cones by taking directional limits; hence the adapted face/tangent hypotheses of [F2] hold for the box pieces in every chart used to fit them.

Collar of the total boundary. If the boundary is empty, the nonrelative construction also gives the relative result. For nonempty boundary build a piecewise smooth collar directly, corners included. Around each boundary point choose a small corner chart and a smooth vector field whose active corner-coordinate components are strictly positive; construct smooth bump functions in slightly larger chart balls, equal to $1$ on the smaller balls and zero near the chart boundary, by applying the explicit smooth cutoff of the secant lemma to squared coordinate distance. Pull them back, extend by zero, and add finitely many interior chart bumps so their sum is positive on $M$. Normalize by that sum and form the weighted sum $V$ of the local inward fields, taking the interior fields to be zero, which is strictly inward along every active local face, because the inward tangent cone is preserved by chart transitions and its strict interior is convex. This gives a smooth field on $M$. In each corner chart its coefficients extend smoothly to a neighbourhood in ${\mathbb{R}}^n$, so [F3] solves the local ODE there and uniqueness makes the solutions agree while in $M$; compactness gives a common positive time for starts on $\mathrm{\partial }M$, and [F3] gives smooth dependence on the initial point. The forward flow $\mathrm{\Phi }$ lies in the interior for small positive times, because every active boundary coordinate has positive derivative.

Fitting inside corner charts. Fit the box pieces and the collar pieces to one another using the supported construction of [F1]. The additional verification in a corner chart is that the specified corner planes are affine, so secant interpolation and the correction extension keep every simplex lying in such a plane inside that same plane; by [F2] sufficiently small strong errors keep each whole box piece in the same orthant, with the same face zero sets, and the derivative zero cones are retained as well: the lower-distance estimate proved in [F2] for the perturbed coordinate, divided by the ray parameter at a zero point, is strictly positive for any tangent direction outside the local zero-face cone. On the zero-face cone the coordinate is identically zero, so its derivative vanishes. The changed maps can therefore be carried back through the original corner chart, without evaluating its inverse outside the chart domain. All other steps of the finite fitting induction are unchanged, and on overlaps local face germs are permuted by the chart transition, so preserving them in the active chart preserves the same geometric faces in every chart; the support margins keep images inside the relatively open charts. Fitting the finitely many pieces successively gives one finite complex and an embedding onto their union, whose corner-face inverse images are subcomplexes, with intersections full after the prescribed barycentric refinement.

The collar is a whole-star corner collar. The map $\mathrm{\partial }M\times [0,ϵ]\to M$, $(y,t)\mapsto \mathrm{\Phi }(y,t)$, is injective for sufficiently small $ϵ$: if two values agree, reverse the common interior orbit by uniqueness; if the times differ, the earlier boundary start would occur at a positive time on an orbit starting at another boundary point, impossible because the forward orbit has already entered the interior and cannot meet the boundary while its inward derivatives remain positive there; equal times give equal starting points. On the compact product it is a homeomorphism onto its image, and that image contains a neighbourhood of $\mathrm{\partial }M$: in a sufficiently small boundary chart, following $-V$ decreases every sufficiently small active face coordinate at a uniformly positive rate until the first one reaches zero, within a time bounded by a constant times the distance to the boundary, so the inverse path is represented as $\mathrm{\Phi }(y,t)$; finitely many boundary charts give a uniform neighbourhood and time. The flow is smooth on each boundary face by [F3], so its composite with $f$ is smooth on each cell of $L\times [0,ϵ]$. At a boundary starting point $y$, write $V_y$ in corner coordinates, with every active component positive. The differential at $(y,t)$ is $D_y{\mathrm{\Phi }}_t(w+s{V}_y)$, where $w=d{f}_y(u)$ is in the boundary differential fan and $s$ is a time tangent direction. The local flow has smooth inverse given by the reverse-time flow, so $D_y{\mathrm{\Phi }}_t$ is invertible; the identity ${\mathrm{\partial }}_t{\mathrm{\Phi }}_t(y)=D_y{\mathrm{\Phi }}_t(V_y)$ follows by differentiating the uniqueness identity ${\mathrm{\Phi }}_{t+a}(y)={\mathrm{\Phi }}_t({\mathrm{\Phi }}_a(y))$ at $a=0$. The map $(w,s)\mapsto w+s{V}_y$ is injective: for its value $v$, necessarily $s={\min }_j{v}_j\mathrm{/}(V_y{)}_j$ over active coordinates, and then $w=v-s{V}_y$. At $t=0$, $s\ge 0$ and these values fill the tangent orthant; at $0<t<ϵ$, $s$ is unrestricted and they fill the full tangent space; at the artificial top, $s\le 0$ gives its corresponding sector. The inverse is continuous by the finite minimum formula. Hence a compatible triangulation of the product cells gives a whole-star smooth collar agreeing with the supplied admissible boundary data.

The union covers $M$. Each initial box has a compact smaller core and these cores cover $M$. In its coordinates a sufficiently small perturbation of its map is $x\mapsto x+e(x)$ on the original box, with $\mathrm{\parallel }e{\mathrm{\parallel }}_{\mathrm{\infty }}$ less than the margin of that core from the artificial box faces and $e_j(x)=0$ on $x_j=0$ for a true corner face. For a target point $y$ in the smaller core, the self-map of the box $x\mapsto {\mathrm{clip}}_{Q_i}(y-e(x))$, clipping each coordinate to its box interval, has a fixed point by [F3], after transporting the box to a ball by radial parametrization from its centre. Clipping at an artificial face cannot occur at a fixed point, because the error is smaller than the core margin; at a true corner face $x_j=0$, fixed-point clipping would give $y_j-e_j(x)\le 0$, hence $y_j=0$ because $e_j(x)=0$ and $y_j\ge 0$, and the equation $x_j+e_j(x)=y_j$ still holds; for unclipped coordinates it holds by definition. Thus the perturbed box still contains its entire core, and choosing the total fitting error below all finitely many such margins makes the union image equal to $M$. Its local differential fans are homeomorphisms onto the tangent orthants: for a target ray $y+tv$ the preimages $x_t$ under the finite union homeomorphism converge to the preimage $b$ of $y$ and lie on a subsequence in one closed simplex containing $b$; full rank gives $\mathrm{\mid }{x}_t-b\mathrm{\mid }\le Ct$ by the first-order expansion and the positive least singular value, and a subsequential limit of $(x_t-b)\mathrm{/}t$ is a tangent-cone vector mapping to $v$, while injectivity of the fan gives continuity of the resulting finite piecewise linear inverse. Hence the union is a finite corner-adapted whole-star smooth triangulation of $M$ with all intersections full.