Reconstruction of Convex Bodies from Congruent Projections

Definitions and mathematical statement

Let $n,k\in{\mathbb{Z}}$ satisfy $n\geq3$ and $2\leq k\leq n-1$. Under the standard convention, a convex body in ${\mathbb{R}}^n$ is a compact convex set with nonempty interior. Let $G(n,k)$ denote the Grassmannian of $k$-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 let $O(H)$ be the group of linear orthogonal transformations of $H$.

Reconstruction question, with an unresolved congruence qualification. In the usual Euclidean meaning of congruent projections, the question is whether, for every pair of convex bodies $K,L\subseteq{\mathbb{R}}^n$, \[ \left[\forall H\in G(n,k)\ \exists U_H\in O(H)\ \exists b_H\in H:\quad P_HK=b_H+U_H(P_HL)\right] \quad\Longrightarrow\quad \left[\exists t\in{\mathbb{R}}^n\ \exists\varepsilon\in\{-1,1\}:\quad K=t+\varepsilon L\right]. \] Here $\varepsilon L=\{\varepsilon x:x\in L\}$, so the conclusion permits a translation of either $L$ or its reflection through the origin. The projection congruence may depend on $H$; the translation $t$ and sign $\varepsilon$ in the conclusion are global.

The source further requires that the congruences be "induced by elements of $O(k,H)$", but does not define this notation or specify whether translations within $H$ are allowed. If it means solely linear orthogonal transformations of $H$, the displayed antecedent must instead be \[ \forall H\in G(n,k)\ \exists U_H\in O(H):\qquad P_HK=U_H(P_HL). \] These antecedents are distinct, and the supplied record does not determine which is intended. Both allow orientation-reversing orthogonal maps; no restriction to direct congruence is stated. Orthogonal projection, linear Grassmannians, and the full-dimensional convex-body convention above are standard interpretations rather than definitions explicitly supplied by the record.

Short English statement

For each $2\leq k\leq n-1$, the question asks whether two convex bodies whose corresponding $k$-dimensional orthogonal projections are congruent must differ by a translation and possibly reflection through the origin. The undefined notation $O(k,H)$ leaves unresolved whether the projection congruences may include translations. The supplied status is partially solved; this is a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003009$, source problem code catalogue entry, title: Reconstruction of Convex Bodies from Congruent Projections. The record gives proposed year 2015 and names no individual 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 supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Alfonseca, Cordier, and Ryabogin on directly congruent projections. https://arxiv.org/abs/1412.2727.
• Supplied literature reference labelled "published reconstruction results". https://doi.org/10.1007/s11856-016-1394-6.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that the general congruent-projection reconstruction question remains unresolved, with affirmative results for symmetric bodies, polytopes, and several additional non-symmetric cases. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The precise hypotheses of those special-case results are not supplied.
• The supplied statement-status label is corrected_verified. Its accompanying qualification says that only safe notation, typography, and TeX-delimiter cleanup was applied, preserving hypotheses, quantifiers, and subquestions. This label does not establish independent mathematical or source verification of this draft.