Why reconstruction from partial face data matters

A convex polytope has both a geometric realization and a finite combinatorial structure: its faces, ordered by inclusion. Reconstruction asks how much of that structure must be known before the rest is forced. Chapter 12 of Branko Grünbaum's Convex Polytopes distinguishes several forms of unambiguity and then proves reconstruction results for general polytopes and for important special classes. The central input is not a coordinate description. It is a lower-dimensional part of the face lattice, such as the graph or a higher skeleton.

This mission groups five established results that address the same recovery problem under different hypotheses. The main theorem reconstructs every d-polytope, for d ≥ 3, from a supplied equivalence of its (d−2)-skeleton. The supporting targets sharpen the amount of data needed for simple polytopes, simplicial polytopes, and zonotopes, and record a polynomial-time version whose output is the vertex–facet incidence relation. Each theorem retains its own quantified polytopes, equivalence, and reconstruction witness. Grouping them does not require a single witness to serve all five regimes.

Face lattices, skeletons, and unambiguity

A d-polytope is represented as the convex hull of a nonempty finite subset of Fin d → ℝ, with affine span equal to the whole ambient space. A face is an exposed face in Mathlib's sense. This convention includes the empty face and the polytope itself. PolytopeFace P is the resulting face poset ordered by set inclusion.

For a natural number k, the k-skeleton face poset SkeletonFace P k contains the empty face and every nonempty exposed face whose affine dimension is at most k. An equivalence of skeletons is an order equivalence, so it preserves and reflects every inclusion relation among the retained faces. Strong unambiguity is represented by demanding an extension of every supplied skeleton equivalence, rather than merely asserting that some full face-lattice equivalence exists. The extension equation requires agreement on every skeleton face.

The graph of a polytope is modeled through its vertices and one-dimensional faces. PolytopeVertex P consists of points whose singleton is exposed. Two distinct vertices are adjacent in polytopeGraph P when they lie in a common one-dimensional exposed face. A source polytope is simple when every vertex has exactly d graph neighbors. A simplicial polytope is one whose facets are simplices. A zonotope is a finite vector sum of closed line segments; full dimension is imposed separately by the d-polytope hypothesis.

Formalization targets

General reconstruction from the (d−2)-skeleton

For d ≥ 3, d-polytopes P and Q, and every supplied order equivalence

$$\varphi:\operatorname{Skel}_{d-2}(P)\simeq_o\operatorname{Skel}_{d-2}(Q),$$

the main target produces an order equivalence of the complete face posets whose restriction is exactly φ. This is the explicit face-lattice reformulation of Theorem 12.3.1. It asserts strong and weak d-unambiguity, but it does not add dimensional unambiguity.

Reconstruction from graphs and half-dimensional skeletons

The Blind–Mani target uses only the 1-skeleton when the source is simple. The target polytope is arbitrary in the same dimension; its simplicity is not assumed separately. The Björner–Edelman–Ziegler target likewise uses the 1-skeleton when the source is a full-dimensional zonotope, without requiring the target to be a zonotope.

Perles's target has a different domain. The source is a simplicial d-polytope, the target is an arbitrary e-polytope, and the input compares their floor(d/2)-skeletons. Its conclusion includes both e=d and extension of the supplied skeleton equivalence. Thus this clause, unlike the other reconstruction clauses, explicitly includes dimensional unambiguity.

Polynomial-time incidence reconstruction

The algorithmic target states that one binary function works uniformly for every d ≥ 3, every vertex count, and every valid labelled d-polytope input. The input contains the complete duplicate-free table of nonempty faces in dimensions at most d−2; the output contains exactly the facets, as labelled vertex sets. Dimension, vertex count, and row count are self-delimiting headers, followed by dense incidence rows. Correctness is required on valid encodings, while the binary function itself is total. ContainmentPolyTime uses Mathlib's Turing-machine polynomial-time predicate together with finite stack alphabets.

Significance

The general theorem says that codimension-two face data already fixes every missing facet and incidence relation. The special-class results show that much smaller data can suffice: only the graph for simple polytopes and zonotopes, and the half-dimensional skeleton for simplicial polytopes. Perles's result also recovers ambient dimension. The algorithmic theorem separates mere uniqueness from effective recovery by requiring one polynomial-time procedure and a concrete vertex–facet output.

Formalizing these results creates reusable interfaces for face-poset reconstruction, graph-based simplicity, zonotopes, and finite incidence encodings. It also records distinctions that informal summaries can blur: extension of every supplied equivalence versus existence of an unrelated isomorphism; combinatorial reconstruction versus dimension recovery; and uniqueness versus polynomial-time computability. The statements are established mathematics, but their Lean theorem bodies remain open and use sorry; this mission makes no proof-completion claim.

Difficulty

Counting faces or matching their dimensions does not reconstruct the face lattice. The conclusion must preserve every inclusion relation and agree with the specific equivalence supplied on the visible skeleton. An arbitrary isomorphism of complete face lattices would be too weak if it failed to extend that map.

For simple polytopes and zonotopes, graph isomorphism alone does not expose higher-dimensional faces directly; the target still asks for all incidences. In the simplicial case, reconstructing the lattice without proving e=d would omit the dimensional-unambiguity clause. For the effective result, a mathematical reconstruction argument is insufficient unless it yields one uniform polynomial-time binary function. Conversely, an efficient procedure on an incomplete or coordinate-dependent input would not meet the stated incidence model.

Formalization scope

All geometric ambient spaces are finite-dimensional real coordinate spaces. Full-dimensional polytopes are nonempty finite convex hulls. Exposed faces include both improper faces. Skeletons retain the empty face separately, avoiding a natural-number encoding of its source dimension −1. Natural subtraction in d−2 is protected by the explicit 3 ≤ d hypothesis wherever that threshold is used.

The graph definition removes loops and symmetrizes adjacency. Simplicity is the exact neighbor-cardinality condition at every vertex. The zonotope definition allows arbitrary segment endpoints and degenerate summands, but the separate full-dimensionality hypothesis rules out a lower-dimensional source. Only the simplicial theorem compares different ambient dimensions and concludes their equality.

The algorithm receives no real coordinates. Its vertex map is injective and has range exactly the exposed singleton faces. The input and output row predicates are biconditionals, so they prohibit missing faces and extraneous rows; Nodup prohibits repetitions. The empty face is implicit, and the output row order is unrestricted. Contributions must not weaken extension to unrelated existence, assume the target belongs to the source's special class, omit Perles's dimension equality, replace complete incidence tables by partial data, or allow the polynomial bound to depend on the individual instance.

Selected references

• Branko Grünbaum, Convex Polytopes, 2nd ed., Graduate Texts in Mathematics 221, Springer, 2003, §§12.1–12.4 and §15.1. DOI: 10.1007/978-1-4613-0019-9.
• Roswitha Blind and Peter Mani-Levitska, Puzzles and polytope isomorphisms, Aequationes Mathematicae 34 (1987), 287–297. DOI: 10.1007/BF01830678.
• Anders Björner, Paul H. Edelman, and Günter M. Ziegler, Hyperplane arrangements with a lattice of regions, Discrete & Computational Geometry 5 (1990), 263–288. DOI: 10.1007/BF02187791.