Normality and Smoothness of Truncated Hadamard Simplices

Definitions and mathematical statement

Let $d$ be a dimension admitting a Hadamard simplex, put $C=[-1,1]^d\subseteq{\mathbb{R}}^d$, and let $\Delta$ be a full-dimensional regular simplex whose $d+1$ vertices are vertices of $C$. Regular means that all its edges have the same Euclidean length. Let $m,n$ be positive odd integers satisfying $n<m<nd$, and define \[P=m\Delta\cap nC,\qquad a=(n,\ldots,n),\qquad M=2{\mathbb{Z}}^d,\qquad\Lambda=a+M=(n+2{\mathbb{Z}})^d.\] Dilations here are about the origin. A lattice polytope in the affine lattice $\Lambda$ means that every vertex of $P$ belongs to $\Lambda$. If this holds, normality is the integer decomposition property relative to this affine lattice: \[\forall k\geq1:\quad kP\cap(ka+M)=\{x_1+\cdots+x_k:x_i\in P\cap\Lambda\}.\] Equivalently, the ordinary lattice polytope $P-a$ is normal in $M$. This uses the correct affine lattice in each dilation, rather than intersecting every $kP$ with the same translate $\Lambda$. Smoothness means that at every vertex the primitive directions of its incident edges form a ${\mathbb{Z}}$-basis of the difference lattice $M$; in particular there must be exactly $d$ such directions. The three questions are: is $P$ always a lattice polytope in $\Lambda$ for the stated data; for which admissible parameters is it normal; and for which is it smooth? The last two use the specified lattice, not the ambient lattice ${\mathbb{Z}}^d$. The record does not determine whether the requested parameter classification is independent of the dimension and the chosen Hadamard simplex. The supplied 2026-08-21 triage labels the problem open, calls the intersection construction incompletely specified, and says the extracted record does not identify a later complete answer, while its source-verification note reports confirmation of the displayed cube, simplex definition, odd parameter range, affine lattice, and three questions.

Short English statement

Intersect a dilated Hadamard simplex with a dilated cube, and ask whether the result is a lattice polytope, when it is normal, and when it is smooth. All lattice notions use the odd-coordinate affine lattice, whose difference lattice is $2{\mathbb{Z}}^d$.

Sources

• Local UnsolvedMath registry, permanent ID 30000813, source problem catalogue entry, Normality and Smoothness of Truncated Hadamard Simplices, proposed year 2007; no individual proposer supplied.
• Supplied citation: Mini-Workshop: Projective Normality of Smooth Toric Varieties (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/39, pp. 2283--2320: https://doi.org/10.4171/owr/2007/39. Supplied literature label: original OWR report.