Normality and Very Ampleness of Lattice Polytopes

Definitions and mathematical statement

A lattice polytope in ${\mathbb{R}}^n$ is the convex hull of a finite nonempty subset of ${\mathbb{Z}}^n$. For such a polytope $P$, write $d=\dim P$, and let $kP=\{kp:p\in P\}$. Use the ambient lattice ${\mathbb{Z}}^n$, including when $P$ has smaller affine dimension. The polytope $P$ is normal if, for every integer $k\geq 1$, \[ kP\cap{\mathbb{Z}}^n =\{u_1+\cdots+u_k:u_1,\ldots,u_k\in P\cap{\mathbb{Z}}^n\}. \] It is nonnormal if this equality fails for some $k\geq 1$.

For a vertex $v$ of $P$, define its tangent cone and its monoid of lattice-point differences by \[ C_v(P)=\{t(p-v):t\geq 0,\ p\in P\},\qquad M_v(P)=\left\{\sum_{u\in P\cap{\mathbb{Z}}^n}m_u(u-v):m_u\in{\mathbb{Z}}_{\geq 0}\right\}. \] Use the convention that $P$ is very ample if \[ M_v(P)=C_v(P)\cap{\mathbb{Z}}^n \quad\text{for every vertex }v\text{ of }P. \] A $d$-dimensional polytope is simple if exactly $d$ edges meet at each vertex. A lattice simplex is the convex hull of $d+1$ affinely independent lattice points. These definitions make explicit the lattice convention used in this draft; the supplied record does not itself define normality or very ampleness.

1. For every dimension $n\geq 1$ and every Euclidean ball $B\subset{\mathbb{R}}^n$ for which $B\cap{\mathbb{Z}}^n\neq\varnothing$, is \[ P_B=\operatorname{conv}(B\cap{\mathbb{Z}}^n) \] normal? Equivalently, must every $x\in kP_B\cap{\mathbb{Z}}^n$, for every integer $k\geq 1$, be a sum of $k$ points of $B\cap{\mathbb{Z}}^n$?
2. Does there exist, in some dimension $n\geq 1$, a very ample nonnormal simple lattice polytope $P\subset{\mathbb{R}}^n$? Does there exist a very ample nonnormal lattice simplex?

The first question has unresolved boundary conventions. The source does not specify whether a ball is $\{x:\|x-c\|_2<r\}$ or $\{x:\|x-c\|_2\leq r\}$, where $c\in{\mathbb{R}}^n$; the question above must be read separately under the intended convention, without asserting that the two versions are equivalent. The center is unrestricted by the source. The source also does not specify whether radius zero is allowed or how to treat a ball containing no lattice points. The nonempty qualification above makes $P_B$ a polytope under the stated definition and leaves the empty case unresolved. No full-dimensionality assumption is supplied; the definitions above allow lower-dimensional polytopes.

Short English statement

The questions ask whether the convex hull of the lattice points in any Euclidean ball is normal, and whether a simple lattice polytope, or even a lattice simplex, can be very ample while failing normality. Ball boundary conventions, empty cases, and the intended lattice convention require review. The supplied record labels these questions open; that status is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003600$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Registry reference: data/unsolvedmath/problems.json. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications (2018), Oberwolfach Reports, pp. 2659--2701, DOI 10.4171/owr/2017/44. https://doi.org/10.4171/owr/2017/44. The record separately gives proposed year 2017; the relationship between these dates has not been verified.
• Supplied literature source: Bruns, questions on normal polytopes. https://www.home.uni-osnabrueck.de/wbruns/brunsw/pdf-article/2-quest-WB.pdf.
• Supplied literature source: Lason--Michalek, nonnormal very ample polytopes. https://arxiv.org/abs/1406.4070.
• Supplied literature source: original OWR report. https://doi.org/10.4171/OWR/2017/44.
• Literature assessment dated exactly 2026-08-21: the supplied curation describes normality for convex hulls of lattice points in Euclidean balls as conjectural, states that very ample nonnormal polytopes are known, and labels the requested simple-polytope and simplex versions unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied registry marks the statement corrected_verified and describes a manual semantic, typographic, and LaTeX review as confirming preservation of the mathematical target. This reports the registry's review claim; it does not establish mathematical correctness or independent source verification for this draft.