Reconstruction from Directly Congruent Convex Projections

Definitions and mathematical statement

Let $n,k\in{\mathbb{Z}}$ satisfy $n\geq3$ and $2\leq k\leq n-1$. Use the customary convention that a convex body in ${\mathbb{R}}^n$ is a compact convex set with nonempty interior. Let $G(n,k)$ denote the set of all $k$-dimensional linear subspaces of ${\mathbb{R}}^n$. For $H\in G(n,k)$, let $P_H:{\mathbb{R}}^n\to H$ be orthogonal projection and define \[ SO(H)=\{R:H\to H: R\text{ is linear and orthogonal},\ \det R=1\}. \] This interprets the source notation $SO(k,H)$.

Interpreting directly congruent projections as congruence by orientation-preserving rigid motions within each projection subspace, the question is whether the following implication holds for every pair of convex bodies $K,L\subseteq{\mathbb{R}}^n$: \[ \left[\forall H\in G(n,k)\ \exists R_H\in SO(H)\ \exists t_H\in H:\quad P_HK=R_H(P_HL)+t_H\right] \Longrightarrow \left[\exists a\in{\mathbb{R}}^n\ \exists\varepsilon\in\{-1,1\}:\quad K=a+\varepsilon L\right]. \] The rotations and projection translations may depend on $H$; the conclusion requires one translation and one sign for the entire bodies. Here $-L=\{-x:x\in L\}$.

There is an unresolved qualification in the supplied wording: directly congruent ordinarily permits translations, and the background refers to a direct-rigid-motion problem, but the statement also says that the congruences are induced by elements of $SO(k,H)$, which consists of linear rotations. If that phrase excludes translations, the alternative hypothesis is $\forall H\in G(n,k)\ \exists R_H\in SO(H):P_HK=R_H(P_HL)$. The supplied record does not explicitly settle this distinction. The convex-body convention above is also not explicitly defined in the source record.

Short English statement

For each $2\leq k\leq n-1$, suppose every pair of corresponding orthogonal $k$-dimensional projections of two convex bodies is directly congruent. Must the bodies differ by a translation, possibly followed by central reflection? Whether translations are allowed in the projection congruences requires source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003010$, title: Reconstruction from Directly Congruent Convex Projections; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record lists proposed year $2015$ and identifies no proposer.
• Supplied report citation: Convex Geometry and its Applications (2016), Oberwolfach Reports, pp. 3179--3269, DOI 10.4171/owr/2015/56. https://doi.org/10.4171/owr/2015/56. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied citation is authoritative report provenance, but its contents have not been independently consulted here.
• Supplied literature reference: Alfonseca, Cordier, and Ryabogin. https://arxiv.org/abs/1412.2727. No paper title or theorem-level citation is supplied.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. The record labels the problem partially solved and reports affirmative results for the special-orthogonal version under restrictions such as two-dimensional projections without rotational symmetries, while describing the full direct-rigid-motion problem as open. The background explicitly calls this a dated literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its accompanying verification note states that only safe notation, typography, and TeX-delimiter cleanup was applied, preserving hypotheses, quantifiers, and subquestions. This is supplied metadata, not independent mathematical verification of this draft.