Lattice Packing Conjecture for Origin-Symmetric Convex Bodies

Definitions and mathematical statement

Let $n$ be a positive integer. Under the standard convention, a convex body $K\subset{\mathbb{R}}^n$ is a compact convex set with nonempty interior, and it is origin-symmetric if $K=-K$. Write $\operatorname{Vol}_n$ for $n$-dimensional Lebesgue measure. A full-rank lattice is a set $L=A{\mathbb{Z}}^n$, where $A\in\operatorname{GL}_n({\mathbb{R}})$; its covolume is $|\det A|$.

The conjecture in the supplied normalized statement is \[ \exists c>0\;\forall n\in{\mathbb{Z}}_{\geq 1}\;\forall K\subset{\mathbb{R}}^n:\quad \left[\begin{gathered} K\text{ is an origin-symmetric convex body},\\ \operatorname{Vol}_n(K)=cn^2 \end{gathered}\right] \Longrightarrow \left[\exists A\in\operatorname{GL}_n({\mathbb{R}}):\ |\det A|=1\ \text{ and }\ (A{\mathbb{Z}}^n)\cap K=\{0\}\right]. \] Thus $c$ must be independent of both $n$ and $K$. With the compact-body convention above, the exclusion applies to the boundary of $K$ as well as its interior.

This formulation follows the supplied normalized statement. The original excerpt refers to an omitted condition "(2)" and an earlier "conclusion", so their precise content cannot be reconstructed from that excerpt alone. In particular, the normalized equality $\operatorname{Vol}_n(K)=cn^2$ and the boundary-inclusive exclusion are retained as supplied; their agreement with the omitted context requires review. The positive-integer dimension range and compact-body convention are standard interpretations of notation not separately defined in the record.

The excerpt presents the assertion as an extension beyond Euclidean balls and ellipsoids and compares it with earlier work of Schmidt under the weaker volume assumption $\operatorname{Vol}_n(K)\leq cn$, citing references [18, 19]. Those references and the complete earlier result are not supplied; the constants in that comparison are not identified with the constant quantified above. The supplied dated assessment labels the general conjecture open and reports the ball and ellipsoid cases as proved. These are supplied source assessments, not independently verified mathematical or literature conclusions.

Short English statement

The supplied conjecture asks for one positive constant, valid in every positive dimension, such that every origin-symmetric convex body of volume $cn^2$ contains no nonzero point of some lattice of covolume one. The stated intersection condition includes boundary points. The original excerpt omits the condition and conclusion it references, so the normalized formulation and its boundary convention require human review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, supplied record with permanent ID $30006506$, source problem code catalogue entry, title "Lattice Packing Conjecture for Origin-Symmetric Convex Bodies", in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The registry gives proposed year 2025 and status open.
• Supplied source citation: Functional Inequalities: Geometric Calculus meets Stochastic Analysis (2026), Oberwolfach Reports, pp. 2917--2994, DOI 10.4171/owr/2025/54. The original excerpt carries page 2939. Report link and supplied literature reference "Oberwolfach Report 2025/54": https://doi.org/10.4171/owr/2025/54. The citation year 2026 and report identifier 2025/54 are reproduced as supplied.
• Supplied literature reference: Lattice packing of spheres using a stochastically evolving ellipsoid, https://link.springer.com/article/10.1007/s00222-026-01412-w. Authors and further bibliographic details are not supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general extension open and reports the Euclidean-ball and ellipsoid cases as proved. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The original excerpt attributes earlier work under $\operatorname{Vol}_n(K)\leq cn$ to Schmidt, with reference labels [18, 19]. Their bibliographic entries are absent from the supplied record.
• The registry marks the statement corrected_verified and reports that the official report was checked and its notation repaired. This records a supplied verification claim; no source visit or independent verification is asserted in this draft.