Spherical Gram simplices and angular links of Euclidean faces

Spherical Gram simplices. Let $n\ge 0$ and let $C=(c_{ij}{)}_{0\le i,j\le n}$ be a real symmetric positive-definite $(n+1)\times (n+1)$ matrix with $c_{ii}=1$ for every $i$ (Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal, A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique). Let $C=L{L}^T$ be its unique Cholesky factorisation with lower triangular $L$ and positive diagonal, and let $u_i\in {\mathbb{R}}^{n+1}$ be the $i$-th row of $L$, regarded as a column vector. Define the positive cone on the vertices and the spherical simplex $K(C):=\{{\sum }_i{\lambda }_i{u}_i:{\lambda }_i\ge 0\},\mathrm{\Sigma }(C):=K(C)\cap S^n,$ where $S^n=\{x\in {\mathbb{R}}^{n+1}:\mathrm{\mid }x\mathrm{\mid }=1\}$ is the unit sphere (Real and complex inner-product spaces and their induced length, The induced length is a norm, Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$); the vertices of $\mathrm{\Sigma }(C)$ are $u_0,\dots ,u_n$. The definition asserts neither that the $u_i$ are unit vectors with the prescribed inner products, nor that $\mathrm{\Sigma }(C)$ lies in a hemisphere, nor any metric statement about it: all of that is proved in Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas ↗.

Angular link of a Euclidean face. Let $C$ be a compact convex polyhedral cell in its Euclidean affine hull with direction space $V$, given by finitely many affine inequalities ${\mathrm{\ell }}_j\ge 0$, and let $F$ be a nonempty face (Finite convex cell complex and linear subdivision). Discard inequalities constant on the affine hull: their constants are nonnegative since $C$ is nonempty, so this does not change $C$. The remaining gradients are nonzero; in dimension zero no inequalities remain. Write $I(F)$ for the set of remaining $j$ with ${\mathrm{\ell }}_j$ vanishing on $F$, $n_j:=\mathrm{\nabla }{\mathrm{\ell }}_j\mathrm{/}\mathrm{\mid }\mathrm{\nabla }{\mathrm{\ell }}_j\mathrm{\mid }$ for the inward unit normal of the defining hyperplane ${\mathrm{\ell }}_j=0$ (a facet normal when that hyperplane cuts out a facet), and $U(F):=\mathrm{span}\{x-y:x,y\in F\}\subseteq V$ for the direction space of $F$, the linear span of the differences of points of $F$, which is $\{0\}$ exactly when $F$ is a vertex. The tangent cone of $C$ at $F$ is $T_{F}C:=\{\xi \in V:⟨\xi ,n_j⟩\ge 0\text{ for all }j\in I(F)\},$ equivalently the closure of $\{\lambda (x-p):x\in C,\lambda \ge 0\}$ for any (hence every) $p$ in the relative interior of $F$, and the normal cone of $F$ in $C$ is $N_{F}C:=T_{F}C\cap U(F{)}^{\perp },U(F{)}^{\perp }=\{\xi \in V:⟨\xi ,u⟩=0\text{ for all }u\in U(F)\};$ the angular link of $F$ in $C$ is the set of unit inward directions normal to $F$, ${\mathrm{Lk}}_C(F):=N_{F}C\cap S(V),S(V)=\{\xi \in V:\mathrm{\mid }\xi \mathrm{\mid }=1\},$ and the angular distance of $\xi ,\eta \in {\mathrm{Lk}}_C(F)$ is $d_{\mathrm{a}\mathrm{n}\mathrm{g}}(\xi ,\eta ):=\arccos ⟨\xi ,\eta ⟩$, the principal inverse cosine (Principal inverse sine and inverse cosine); equivalently it is the intrinsic path metric of the round unit sphere restricted to the link, a metric of diameter at most $\pi$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric; proved in Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas ↗(v), using that $N_{F}C$ is a convex cone). For a point $p$ in the relative interior of $F$ the set of all unit inward directions at $p$ is ${\mathrm{Lk}}_C(p):=T_{F}C\cap S(V)$; when $\dim F=k$ it is the spherical join $S^{k-1}\ast {\mathrm{Lk}}_C(F)$ of the round unit sphere of $U(F)$ with the link of $F$, and for a vertex $F$ it coincides with ${\mathrm{Lk}}_C(F)$ (proved in The cone and join metrics and the local product chart of a polyhedral gluing(4)).

For an isometric polyhedral gluing $X$ with cells $C_p$ (Abstract isometric polyhedral gluings and the chain metric) the links ${\mathrm{Lk}}_{C_p}(F)$ of the cells containing a face $F$ are identified along the isometries induced by the gluing maps $h_{p,q}$, giving the angular link ${\mathrm{Lk}}_X(F)$ with the componentwise intrinsic path distance of the cells, extended by the auxiliary value $+\mathrm{\infty }$ between components (The angular path metric, the Euclidean cone and spherical joins(1)); the links ${\mathrm{Lk}}_{C_p}(p)$ of the point $p$ glue in the same way to ${\mathrm{Lk}}_X(p)$. Its link cells are the unit normal direction sets in the cofaces $G>F$, with the induced face incidences. When $X$ is simplicial, the correspondence $G\mapsto G\setminus F$ identifies these cells with the nonempty simplices of the existing combinatorial link (Subcomplexes, closures, stars, and links in a simplicial complex); for general polyhedral cells this is a polyhedral face link, not an abstract simplicial link without subdivision.

• Sign convention of the normals. The normals are inward: each $n_j$ points into $C$ along the increasing direction of ${\mathrm{\ell }}_j$, so the tangent cone at a face is cut out by $⟨\xi ,n_j⟩\ge 0$. With outward normals every inequality would be reversed. The two descriptions of $T_{F}C$ displayed above are proved to agree in Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas ↗(v), together with the facts that $T_{F}C$ is independent of the chosen point in the relative interior of $F$ and that the angular distance is the intrinsic metric of the link.
• What is deferred to the justifier. The unit-norm and inner-product properties of the $u_i$ and the uniqueness of $\mathrm{\Sigma }(C)$ up to an isometry of $S^n$ (clause (i)), the hemisphere and radial coordinates (clause (ii)), the metric axioms and intrinsic description of $d_{\mathrm{a}\mathrm{n}\mathrm{g}}$ (clause (v)) and the Schur-complement formula for vertex links (clause (iv)) are all conclusions of Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas ↗; nothing beyond the construction is asserted here.
• Why the vertices are not assumed to be unit vectors. The Cholesky factor of $C$ is used only as a convenient ambient realisation; that its rows are unit vectors with inner products $c_{ij}$ is the content of clause (i) of the justifier, not part of the construction.
• Face link versus point link. The angular link ${\mathrm{Lk}}_C(F)$ of a face collects only the directions normal to $F$ and has dimension ${\mathrm{codim}}_{C}F-1$, with the empty link in codimension zero. In the simplicial case its cells are those of the combinatorial link of $F$. The larger set ${\mathrm{Lk}}_C(p)=T_{F}C\cap S(V)$ of all inward directions at a point $p$ in the relative interior of $F$ is the join $S^{k-1}\ast {\mathrm{Lk}}_C(F)$, and the two notions coincide exactly when $F$ is a vertex. The cone charts of The cone and join metrics and the local product chart of a polyhedral gluing(4) use ${\mathrm{Lk}}_X(p)$, while the Schur-complement and metric-flag computations use the face link ${\mathrm{Lk}}_X(F)$.