Volume Bounds for Symmetric Lattice Polytopes

Definitions and mathematical statement

Symmetric lattice-polytope volume conjecture. For a full-dimensional origin-symmetric lattice polytope $P\subset{\mathbb{R}}^n$, $n\ge1$, the proposed inequality is\[\operatorname{vol}_n(P)\ge\frac{2^{n-1}}{n!}\left(G(\operatorname{int}P)+1\right),\qquad G(\operatorname{int}P)=\#(\operatorname{int}P\cap{\mathbb{Z}}^n).\] Here a lattice polytope is the convex hull of finitely many lattice points, origin symmetry means $P=-P$, and $\operatorname{int}$ is ordinary ambient interior. Volume is Euclidean Lebesgue measure normalized so that a fundamental cell of ${\mathbb{Z}}^n$ has volume one; it is not normalized lattice volume multiplied by $n!$. The plus one and coefficient $2^{n-1}/n!$ are retained exactly.

Full dimensionality is not stated explicitly in the source but is a necessary qualification for this ambient-volume formulation. If lower-dimensional polytopes are included, both ambient volume and the count of ambient-interior points are zero, while the right side remains positive. A relative-dimensional variant would need dimension, relative interior, and induced-lattice volume specified separately, and is not silently substituted. In the full-dimensional case the origin is an interior lattice point by symmetry, so the count is not a boundary-inclusive lattice-point enumerator. No smoothness, simplicity, or upper bound on the number of interior lattice points is imposed.

The supplied triage checked 2026-08-21 labels the conjecture open and reports no located complete resolution, citing only the report. It supplies no exact later bound or dimensional status comparison. The universal inequality is preserved with its full-dimensional qualification and without an independently asserted present-day status.

Short English statement

The conjecture bounds Euclidean volume below by the stated multiple of the interior lattice-point count plus one. Ambient full dimensionality is essential; using lower-dimensional polytopes literally would make the formula false. Boundary points and normalized lattice volume are not substituted.

Sources

• Local UnsolvedMath registry, permanent ID 30000638, code catalogue entry; Volume Bounds for Symmetric Lattice Polytopes, Oberwolfach Reports Open Problems.
• Konvexgeometrie (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/56, pp. 3321--3396. https://doi.org/10.4171/owr/2006/56. Proposed year and year tag: 2006; no proposer named.
• The same full report is the sole supplied literature source; no separate authoritative conjecture or later theorem citation is supplied.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field records notation normalization, not an explicit dimensional convention. Verify dated claims independently.