Even-Dimensional Stress-Space Reconstruction

Definitions and mathematical statement

Let $k\in{\mathbb{Z}}$ with $k\geq2$, and put $d=2k$. A simplicial $d$-polytope is a full-dimensional convex polytope $P\subset{\mathbb{R}}^d$ whose proper faces are simplices. Its natural embedding places each vertex at its actual position in ${\mathbb{R}}^d$. Write $V(P)$ for its vertex set and $\Delta(P)$ for its boundary complex. A missing face is a nonempty subset $F\subseteq V(P)$ such that $F\notin\Delta(P)$ but every proper subset of $F$ belongs to $\Delta(P)$; its dimension is $|F|-1$. The supplied hypothesis is \[ F\text{ a missing face of }\Delta(P)\quad\Longrightarrow\quad |F|-1<d-k+1=k+1. \] Thus every missing face has dimension at most $k$.

Write $\mathcal S_k^{\mathrm{aff}}(P)$ for the space termed the space of affine $k$-stresses in the supplied record. The intended assertion is that, for every such $P$, this stress-space data determines $P$ up to affine equivalence. Here $P$ and $Q$ are affinely equivalent if there exist $A\in\operatorname{GL}_d({\mathbb{R}})$ and $b\in{\mathbb{R}}^d$ such that $Q=\{Ax+b:x\in P\}$.

For a comparison formulation within the stated class, let $P,Q\subset{\mathbb{R}}^{2k}$ satisfy the preceding hypotheses and let $\phi:V(P)\to V(Q)$ be a bijection. The intended reconstruction implication is that identification of their affine $k$-stress data through $\phi$ forces affine equivalence of $P$ and $Q$. However, the supplied record does not define affine $k$-stresses, their ambient coordinate space, or the data retained when two stress spaces are identified. It also does not specify whether comparison polytopes must satisfy the missing-face hypothesis or whether the resulting affine equivalence must realize $\phi$. Consequently, this comparison formulation is conditional on resolving those conventions; an abstract vector-space isomorphism is not silently adopted as the meaning of identical stress data.

Short English statement

For each integer $k\geq2$, the supplied conjectural assertion says that a naturally embedded simplicial $2k$-polytope with no missing faces of dimension at least $k+1$ can be reconstructed up to affine equivalence from its affine $k$-stress space. The precise stress-space data and comparison conventions require clarification from the cited source.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30005717$, title Even-Dimensional Stress-Space Reconstruction, exact source problem code catalogue entry. The registry records proposed year $2023$.
• Geometric, Algebraic and Topological Combinatorics (2024), Oberwolfach Reports, pp. 3249--3318, DOI 10.4171/owr/2023/58: https://doi.org/10.4171/owr/2023/58. The supplied original excerpt carries page number $3303$, attributes the surrounding conjecture to Kalai through reference [4], and states that the $d=2k\geq4$ case remains open. The bibliographic identity of reference [4] is not supplied.
• Supplied literature source labelled "Novik--Zheng, affine stresses and reconstruction": https://arxiv.org/abs/2208.06693.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the broader problem "partially solved", reports that the $k=1$ and $k=\lfloor d/2\rfloor+1$ boundary cases are settled, and describes the even-dimensional $d=2k\geq4$ case as still open. This is supplied dated curation, not an independently verified literature assessment; the boundary cases concern the broader conjecture rather than additional cases within the restricted parameters above.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft.