# Normal polytopes and ellipsoids

Joseph Gubeladze  Department of Mathematics
San Francisco State University
1600 Holloway Ave.
San Francisco, CA 94132, USA<EMAIL_ADDRESS>
###### Abstract.

We show that: (1) unimodular simplices in a lattice 3-polytope cover a
neighborhood of the boundary of the polytope if and only if the polytope is
very ample, (2) the convex hull of lattice points in every ellipsoid in
$\mathbb{R}^{3}$ has a unimodular cover, and (3) for every $d\geq 5$, there
are ellipsoids in $\mathbb{R}^{d}$, such that the convex hulls of the lattice
points in these ellipsoids are not even normal. Part (c) answers a question of
Bruns, Michałek, and the author.

###### Key words and phrases:

Normal polytope, Unimodular cover, Ellipsoid

###### 2010 Mathematics Subject Classification:

Primary 52B20; Secondary 11H06

##  1\. Introduction

###  1.1. Main result

A convex polytope $P\subset\mathbb{R}^{d}$ is _normal_ if it is _lattice_ ,
i.e., has vertices in $\mathbb{Z}^{d}$, and satisfies the condition

$\forall c\in\mathbb{N}\quad\forall x\in(cP)\cap\mathbb{Z}^{d}\quad\exists
x_{1},\ldots,x_{c}\in P\cap\mathbb{Z}^{d}\qquad x_{1}+\cdots+x_{c}=x.$

A necessary condition for $P$ to be normal is that the subgroup

$\operatorname{gp}(P):=\sum_{x,y\in
P\cap\mathbb{Z}^{d}}\mathbb{Z}(x-y)\subset\mathbb{Z}^{d}$

must be a direct summand. Also, a face of a normal polytope is normal.

Normality is a central notion in toric geometry and combinatorial commutative
algebra [7]. A weaker condition for lattice polytopes is _very ample_ ; see
Section 1.2 for the definition. Normal polytopes define projectively normal
embeddings of toric varieties whereas very ample polytopes correspond to
normal projective varieties [3, Proposition 2.1].

A sufficient condition for a lattice polytope $P$ to be normal is the
existence of a _unimodular cover,_ which means that $P$ is a union of
unimodular simplices. Unimodular covers play an important role in integer
programming through their connection to the _Integral Carathéodory Property_
[8, 12, 15].

There exist normal polytopes in dimensions $\geq 5$ without unimodular cover
[6]. It is believed that all normal 3-polytopes have unimodular cover. But
progress in this direction is scarce. Recent works [4, 11] show that all
lattice 3-dimensional parallelepipeds and centrally symmetric 3-polytopes with
unimodular corners have unimodular cover.

The normality of the convex hull of lattice points in an ellipsoid naturally
comes up in [9]. We consider general ellipsoids, neither centered at $0$ nor
aligned with the coordinate axes. According to [9, Theorem 6.5(c)], the convex
hull of the lattice points in any ellipsoid $E\subset\mathbb{R}^{3}$ is
normal. [9, Question 7.2(b)] asks whether this result extends to higher
dimensional ellipsoids.

Here we prove the following

Theorem. _Let $P\subset\mathbb{R}^{3}$ be a lattice polytope,
$E\subset\mathbb{R}^{d}$ an ellipsoid, and $P(E)$ the convex hull of the
lattice points in $E$._

  1. (a)

_The unimodular simplices in_ $P$ cover a neighborhood of the boundary
$\partial P$ in $P$ if and only if $P$ is very ample.

  2. (b)

_If_ $d=3$ then the polytope $P(E)$ is covered by unimodular simplices.

  3. (c)

_For every_ $d\geq 6$, there exists $E$ such that
$\operatorname{gp}(P(E))=\mathbb{Z}^{d}$ and $P(E)$ is not normal.

If in (c) we drop the condition $\operatorname{gp}(P(E))=\mathbb{Z}^{d}$, then
ellipsoids $E\subset\mathbb{R}^{d}$ with $P(E)$ non-normal already exist for
$d=5$; see Remark 4.3.

###  1.2. Preliminaries

$\mathbb{Z}_{+}$ and $\mathbb{R}_{+}$ denote the sets of non-negative integers
and reals, respectively.

The convex hull of a set $X\subset\mathbb{R}^{d}$ is denoted by
$\operatorname{conv}(X)$. The relative interior of a convex set
$X\subset\mathbb{R}^{d}$ is denoted by $\operatorname{int}X$. The boundary of
$X$ is denoted by $\partial X=X\setminus\operatorname{int}X$.

Polytopes are assumed to be convex. For a polytope $P\subset\mathbb{R}^{d}$,
its vertex set is denoted by $\operatorname{vert}(P)$.

A lattice $n$-simplex
$\Delta=\operatorname{conv}(x_{0},\ldots,x_{n})\subset\mathbb{R}^{d}$ is
_unimodular_ if $\\{x_{1}-x_{0},\ldots,x_{n}-x_{0}\\}$ is a part of a basis of
$\mathbb{Z}^{d}$.

A _unimodular pyramid_ over a lattice polytope $Q$ is a lattice polytope
$P=\operatorname{conv}(v,Q)$, where the point $v$ is not in the affine hull of
$Q$ and the lattice height of $v$ above $Q$ inside the affine hull of $P$
equals 1.

Cones $C$ are assumed to be _pointed, rational,_ and _finitely generated,_
which means $C=\mathbb{R}_{+}x_{1}+\cdots+\mathbb{R}_{+}x_{k}$, where
$x_{1},\ldots,x_{k}\in\mathbb{Z}^{d}$ and $C$ does not contain a nonzero
linear subspace. For a cone $C\subset\mathbb{R}^{d}$, the smallest generating
set of the additive submonoid $C\cap\mathbb{Z}^{d}\subset\mathbb{Z}^{d}$
consists of the indecomposable elements of this monoid. This is a finite set,
called the _Hilbert basis_ of $C$ and denoted by $\operatorname{Hilb}(C)$. See
[7, Chapter 2] for a detailed discussion on Hilbert bases. For a lattice
polytope $P\subset\mathbb{R}^{d}$, we have the inclusion of finite subsets of
$\mathbb{Z}^{d+1}$:

$\displaystyle\big{(}P\cap\mathbb{Z}^{d},1\big{)}\subset\operatorname{Hilb}(\mathbb{R}_{+}(P,1)).$

This inclusion is an equality if and only if $P$ is normal.

A lattice polytope $P$ is _very ample_ if
$\operatorname{Hilb}(\mathbb{R}_{+}(P-v))\subset P-v$ for every vertex
$v\in\operatorname{vert}(P)$. All normal polytopes are very ample, but already
in dimension $3$ there are very ample non-normal polytopes [7, Exercise 2.24].
For a detailed analysis of the discrepancy between the two properties see [3].

For a cone $C\subset\mathbb{R}^{d}$, we say that $C$ has a _unimodular Hilbert
triangulation (cover)_ if $C$ can be triangulated (resp., covered) by cones of
the form $\mathbb{R}_{+}x_{1}+\cdots+\mathbb{R}x_{n}$, where
$\\{x_{1},\ldots,x_{n}\\}$ is a part of a basis of $\mathbb{Z}^{d}$ as well as
of $\operatorname{Hilb}(C)$.

An _ellipsoid_ $E\subset\mathbb{R}^{d}$ is a set of the form

$\displaystyle\big{\\{}x\in\mathbb{R}^{d}\ |\
(l_{1}(x)-a_{1})^{2}+\cdots+(l_{d}(x)-a_{d})^{2}=1\big{\\}}\subset\mathbb{R}^{d},$

where $l_{1},\ldots,l_{d}$ is a full-rank system of real linear forms and
$a_{1},\ldots,a_{d}\in\mathbb{R}^{d}$.

For a lattice polytope $P$, the union of unimodular simplices in $P$ will be
denoted by ${\operatorname{U}}(P)$.

##  2\. Unimodular covers close to the boundary

The following result of Sebő was later rediscovered in [1, 5] in a refined
form in the context of toric varieties.

######  Theorem 2.1.

([16]) Every 3-dimensional cone $C$ has a unimodular Hilbert triangulation.

_Notice._ There exist 4-dimensional cones without unimodular Hilbert
triangulation [5] and it is not known whether all 4- and 5-dimensional cones
have unimodular Hilbert cover. According to [6], in all dimensions $\geq 6$
there are cones without unimodular Hilbert cover.

If $P\subset\mathbb{R}^{3}$ is very ample, then by Theorem 2.1, for every
$v\in\operatorname{vert}(P)$, the cone $\mathbb{R}_{+}(P-v)$ has a unimodular
Hilbert triangulation:

$\mathbb{R}_{+}(P-v)=\bigcup_{T(v)}C_{t},$

where $T(v)$ is a finite index set, depending on $v$. In particular, the
following unimodular simplices form a neighborhood of $v$ in $P$:

$\Delta_{v,t}=\operatorname{conv}(\operatorname{Hilb}(C_{t}),0)+v,\quad t\in
T(v).$

Also, lattice polygons have unimodular triangulation [7, Corollary 2.54].
Therefore, the following lemma completes the proof of Theorem (a):

######  Lemma 2.2.

For a lattice polytope $P$ of an arbitrary dimension, the following conditions
are equivalent:

  1. (a)

${\operatorname{U}}(P)$ is a neighborhood of $\partial P$ within $P$;

  2. (b)

${\operatorname{U}}(P)$ is a neighborhood within $P$ of every vertex of $P$
and $\partial P\subset{\operatorname{U}}(P)$.

###### Proof.

The implication (a)$\Longrightarrow$(b) is obvious.

For the opposite implication, let:

  1. $\centerdot$

$x\in\partial P$;

  2. $\centerdot$

$F$ be the minimal face of $P$ containing $x$;

  3. $\centerdot$

$v\in\operatorname{vert}(F)$;

  4. $\centerdot$

$T_{F}$ be a unimodular cover of $F$ with $\dim(F)$-simplices, contained in
$F$;

  5. $\centerdot$

$T_{v}$ be a unimodular cover of a neighbourhood of $v$ in $P$;

  6. $\centerdot$

$T_{v,F}$ be the sub-family of $T_{v}$, consisting of simplices that have a
$\dim(F)$\- dimensional intersection with $F$;

  7. $\centerdot$

$T_{v}/F$ be the collection of faces of simplices in $T_{v,F}$, opposite to
$F$ (that is, from each simplex in $T_{v,F}$ remove the $\dim(F)+1$ vertices
that lie in $F$, so that one is left with a $(\dim(P)-\dim(F))$-simplex).

Then, the collection of $\operatorname{conv}(T_{v}/F,T_{F})$ covers a
neighbourhood of $x$ in $P$ and consists of unimodular simplices. ∎

##  3\. Unimodular covers inside ellipsoids

###  3.1. Proof of Theorem (b)

The set of normal polytopes $P\subset\mathbb{R}^{d}$ carries a poset
structure, where the order is generated by the elementary relation

$P\leq Q\ \text{if}\ P\subset Q\ \text{and}\
\\#(Q\cap\mathbb{Z}^{d})=\\#(P\cap\mathbb{Z}^{d})+1.$

In [9] this poset is denoted by ${\operatorname{NPol}}(d)$. The trivial
minimal elements of ${\operatorname{NPol}}(d)$ are the singletons from
$\mathbb{Z}^{d}$. It is known that ${\operatorname{NPol}}(d)$ has nontrivial
minimal elements for $d\geq 4$ [7, Exercise 2.27] and the first maximal
elements for $d=4,5$ were found in [9]. It is possible that
${\operatorname{NPol}}(d)$ has isolated elements for some $d$.

Computer searches so far have found neither maximal nor nontrivial minimal
elements in ${\operatorname{NPol}}(3)$ [9]. The next lemma is yet another
evidence that all normal 3-polytopes have unimodular cover.

######  Lemma 3.1.

Let $P$ be a normal 3-polytope. If $*\leq P$ in ${\operatorname{NPol}}(3)$ for
a singleton $*\in\mathbb{Z}^{3}$ then $P={\operatorname{U}}(P)$.

###### Proof.

If $Q\leq P$ is an elementary relation in ${\operatorname{NPol}}(d)$ and $\dim
Q<\dim P$ then $P$ is a unimodular pyramid over $Q$. In this case every full-
dimensional unimodular simplex $\Delta\subset P$ is the unimodular pyramid
over a unimodular simplex in $Q$ and with the same apex as $P$. On the other
hand, lattice segments and polygons are unimodularly triangulable. Therefore,
it is enough to show that a polytope $P\in{\operatorname{NPol}}(3)$ has a
unimodular cover if there is a 3-polytope $Q\in{\operatorname{NPol}}(3)$, such
that $Q$ has a unimodular cover and $Q\leq P$ is an elementary relation in
${\operatorname{NPol}}(3)$. Assume $\\{v\\}=\operatorname{vert}(P)\setminus
Q$. By Theorem (a) we have the inclusion
$P\setminus{\operatorname{U}}(P)\subset Q$. Since $Q={\operatorname{U}}(Q)$ we
have $P={\operatorname{U}}(P)$. ∎

Call a subset $\mathcal{E}\subset\mathbb{Z}^{d}$ _ellipsoidal_ and a point
$v\in\mathcal{E}$ _extremal_ if there is an ellipsoid
$E\subset\mathbb{R}^{d}$, such that
$\mathcal{E}=\operatorname{conv}(E)\cap\mathbb{Z}^{d}$ and $v\in E$.

######  Lemma 3.2.

Let $\mathcal{E}\subset\mathbb{R}^{d}$ be an ellipsoidal set. Then
$\mathcal{E}$ has an extremal point and $\mathcal{E}\setminus\\{v\\}$ is also
ellipsoidal for every extremal point $v\in\mathcal{E}$.

###### Proof.

Let $\mathcal{E}=\operatorname{conv}(E)\cap\mathbb{Z}^{d}$ for an ellipsoid
$E\subset\mathbb{R}^{d}$. Applying an appropriate homothetic contraction,
centered at the center of $E$, we can always achieve $\mathcal{E}\cap
E\not=\emptyset$. In particular, $\mathcal{E}$ has an extremal point. For
$v\in\mathcal{E}\cap E$, after changing $E$ to its homothetic image with
factor $(1+\varepsilon)$ and centered at $v$, where $\varepsilon$ is a
sufficiently small positive real number, we can further assume
$\mathcal{E}\cap E=\\{v\\}$. Finally, applying a parallel translation to $E$
by $\delta(z-v)$, where $z$ is the center of $E$ and $\delta>0$ is a
sufficiently small real number, we achieve
$\operatorname{conv}(E)\cap\mathbb{Z}^{d}=\mathcal{E}\setminus\\{v\\}$. ∎

Next we complete the proof of Theorem (b). It follows from Lemma 3.2 that, for
any natural number $d$ and an ellipsoidal set
$\mathcal{E}\subset\mathbb{Z}^{d}$, there is a descending sequence of
ellipsoidal sets of the form

$\displaystyle\mathcal{E}=\mathcal{E}_{k}\supset\mathcal{E}_{k-1}\supset\ldots\supset\mathcal{E}_{1},\
\text{with}\ \\#\mathcal{E}_{i}=i\ \text{for}\ i=1,\ldots,k.$

By [9, Theorem 6.5(c)], for $d=3$, the $\operatorname{conv}(\mathcal{E}_{i})$
are normal polytopes. Therefore, $*\leq\operatorname{conv}(\mathcal{E})$ in
${\operatorname{NPol}}(3)$ for some $*\in\mathbb{Z}^{3}$. Thus Lemma 3.1
applies.∎

###  3.2. Alternative algorithmic proof in symmetric case

For the ellipsoids $E$ with center in $\frac{1}{2}\mathbb{Z}^{3}$, there is a
different proof of Theorem (b). It yields a simple algorithm for constructing
a unimodular cover of $P(E)$.

Instead of Theorem 2.1 and [9, Theorem 6.5] this approach uses Johnson’s 1916
_Circle Theorem_ [13, 14]. We only need Johnson’s theorem to derive the
following fact, which does not extend to higher dimensions: for any lattice
$\Lambda\subset\mathbb{R}^{2}$ and any ellipse
$E^{\prime}\subset\mathbb{R}^{2}$, such that $\operatorname{conv}(E^{\prime})$
contains a triangle with vertices in $\Lambda$, every parallel translate
$\operatorname{conv}(E^{\prime})+v$, where $v\in\mathbb{R}^{2}$, meets
$\Lambda$.

Assume an ellipsoid $E\subset\mathbb{R}^{3}$ has center in
$\frac{1}{2}\mathbb{Z}^{3}$ and $\dim(P(E))=3$ (notation as in the theorem).
Assume ${\operatorname{U}}(P(E))\subsetneqq P(E)$. Because $\partial P(E)$ is
triangulated by unimodular triangles, there is a unimodular triangle $T\subset
P(E)$, not necessarily in $\partial P(E)$, and a point
$x\in\operatorname{int}T$, such that the points in $[0,x]$, sufficiently close
to $x$, are not in ${\operatorname{U}}(P(E))$. For the plane, parallel to $T$
on lattice height 1 above $T$ and on the same side as $0$, the intersection
$E^{\prime}=\operatorname{conv}(E)\cap H$ is at least as large as the
intersection of $\operatorname{conv}(E)$ with the affine hull of $T$: a
consequence of the fact that $P(E)\cap\mathbb{Z}^{3}$ is symmetric relative to
the center of $E$. The mentioned consequence of Johnson’s theorem implies that
$\operatorname{conv}(E^{\prime})$ contains a point $z\in\mathbb{Z}^{3}$. In
particular, all points in $[x,0]$, sufficiently close to $x$ are in the
unimodular simplex $\operatorname{conv}(T,z)\subset P(E)$, a contradiction.

##  4\. High dimensional ellipsoids

For a lattice $\Lambda\subset\mathbb{R}^{d}$, define a _$\Lambda$ -polytope_
as a polytope $P\subset\mathbb{R}^{d}$ with
$\operatorname{vert}(P)\subset\Lambda$. Using $\Lambda$ as the lattice of
reference instead of $\mathbb{Z}^{d}$, one similarly defines _$\Lambda$
-normal polytopes_ and _$\Lambda$ -ellipsoidal sets_.

Consider the lattice
$\Lambda(d)=\mathbb{Z}^{d}+\mathbb{Z}\left(\frac{1}{2},\ldots,\frac{1}{2}\right)\subset\mathbb{R}^{d}$.
We have $\big{[}\mathbb{Z}^{d}:\Lambda(d)\big{]}=2$. Consider the
$\Lambda(d)$-polytope
$P(d)=\operatorname{conv}\big{(}\operatorname{B}(d)\cap\Lambda(d)\big{)}$,
where $\operatorname{B}(d)=\big{\\{}(\xi_{1},\ldots,\xi_{d})\ \big{|}\
\sum_{i=1}^{d}\big{(}\xi_{i}-\frac{1}{2}\big{)}^{2}\leq\frac{d}{4}\big{\\}}\subset\mathbb{R}^{d}$,
i.e., $\partial(\operatorname{B}(d))$ is the circumscribed sphere for the cube
$[0,1]^{d}$.

Consider the $d$-dimensional $\Lambda(d)$-polytope and the $(d-1)$-dimensional
$\Lambda(d)$-simplex:

$\displaystyle
Q(d)=\operatorname{conv}\big{(}(P(d)\cap\Lambda(d))\setminus\\{\mathbf{e}_{1}+\cdots+\mathbf{e}_{d}\\}\big{)},$
$\displaystyle\Delta(d-1)=\operatorname{conv}\left(\mathbf{e}_{1}+\cdots+\mathbf{e}_{i-1}+\mathbf{e}_{i+1}+\cdots+\mathbf{e}_{d}\
|\ i=1,\ldots,d\right),$

where $\mathbf{e}_{1},\ldots,\mathbf{e}_{d}\in\mathbb{R}^{d}$ are the standard
basic vectors.

_Notice._ Although $P(d)\cap\mathbb{Z}^{d}=\\{0,1\\}^{d}$ for all $d$, yet
$[0,1]^{d}\subsetneqq P(d)$ for all $d\geq 4$. In fact,
$\big{(}\frac{1}{2},\ldots,\frac{1}{2}\big{)}+k\mathbf{e}_{i}\in
P(d)\cap\Lambda(d)$ for $1\leq i\leq d$ and
$-\left\lceil\frac{\sqrt{d}}{2}\right\rceil\leq
k\leq\left\lfloor\frac{\sqrt{d}}{2}\right\rfloor$.

######  Lemma 4.1.

If $d\geq 5$ then $\Delta(d-1)$ is a facet of $Q(d)$ and
$\Delta(d-1)\cap\Lambda(d)=\operatorname{vert}(\Delta(d-1))$.

###### Proof.

Assume $x=(\xi_{1},\ldots,\xi_{d})\in P(d)\cap\Lambda(d)$ satisfies
$\xi_{1}+\cdots+\xi_{d}\geq d-1$. We claim that there are only two
possibilities: either $x=\mathbf{e}_{1}+\cdots+\mathbf{e}_{d}$ or
$x=\mathbf{e}_{1}+\cdots+\mathbf{e}_{i-1}+\mathbf{e}_{i+1}+\cdots+\mathbf{e}_{d}$
for some index $i$. Since $P(d)\cap\mathbb{Z}^{d}=\\{0,1\\}^{d}$, only the
case $x\in\left(\frac{1}{2},\ldots,\frac{1}{2}\right)+\mathbb{Z}^{d}$ needs to
be ruled out. Assume $\xi_{i}=\frac{1}{2}+a_{i}$ for some integers $a_{i}$,
where $i=1,\ldots,d$. Then we have the inequalities

$\displaystyle\sum_{i=1}^{d}a_{i}^{2}\leq\frac{d}{4}\quad\text{and}\quad\sum_{i=1}^{d}a_{i}\geq\frac{d}{2}-1.$

Since the $a_{i}$ are integers we have $\frac{d}{4}\geq\frac{d}{2}-1$, a
contradiction because $d\geq 5$. ∎

######  Lemma 4.2.

For every even natural number $d\geq 6$, there exists a point in
$\big{(}\frac{d}{2}\cdot Q(d)\big{)}\cap\Lambda(d)$ which does not have a
representation of the form $x_{1}+\cdots+x_{\frac{d}{2}}$ with
$x_{1},\ldots,x_{\frac{d}{2}}\in Q(d)\cap\Lambda(d)$. In particular, $Q(d)$ is
not $\Lambda(d)$-normal.

###### Proof.

Consider the baricenter
$\beta(d)=\frac{d-1}{d}\cdot(\mathbf{e}_{1}+\cdots+\mathbf{e}_{d})$ of
$\Delta(d-1)$. The point $\frac{d}{2}\cdot\beta(d)$ is the baricenter of the
dilated simplex $\frac{d}{2}\cdot\Delta(d-1)$ and, simultaneously, a point in
$\Lambda(d)$. Assume $\frac{d}{2}\cdot\beta=x_{1}+\cdots+x_{\frac{d}{2}}$ for
some $x_{1},\ldots,x_{\frac{d}{2}}\in Q(d)\cap\Lambda(d)$. Lemma 4.1 implies
$x_{1},\ldots,x_{\frac{d}{2}}\in\Delta(d-1)\cap\Lambda(d)=\operatorname{vert}(\Delta(d-1))$.
But this is not possible because the dilated $(d-1)$-simplex $c\Delta(d-1)$
has an interior point of the form $z_{1}+\cdots+z_{c}$ with
$z_{1},\ldots,z_{c}\in\operatorname{vert}(\Delta(d-1))$ only if $c\geq d$. ∎

###### Proof of Theorem (c).

Since
$\mathbf{e}_{1},\ldots,\mathbf{e}_{d},(\frac{1}{2},\ldots,\frac{1}{2})\in
Q(d)$ we have the equality $\operatorname{gp}(Q(d))=\Lambda(d)$. By Lemmas 3.2
and 4.1, the set $Q(d)\cap\Lambda(d)$ is $\Lambda(d)$-ellipsoidal for $d\geq
5$. By applying a linear transformation, mapping $\Lambda(d)$ isomorphically
to $\mathbb{Z}^{d}$, Lemma 4.2 already implies Theorem (c) for $d$ even.

One involves all dimensions $d\geq 6$ by observing that (i) if
$\mathcal{E}\subset\mathbb{R}^{d}$ is an ellipsoidal set then
$\mathcal{E}\times\\{0,1\\}\subset\mathbb{R}^{d+1}$ is also ellipsoidal and
(ii) the normality of $\operatorname{conv}(\mathcal{E}\times\\{0,1\\})$
implies that of $\operatorname{conv}(\mathcal{E})$. While (ii) is
straightforward, for (i) one applies an appropriate affine transformation to
achieve $\mathcal{E}=\operatorname{conv}(S^{d-1})\cap\Lambda$, where
$S^{d-1}\subset\mathbb{R}^{d}$ is the unit sphere, and
$\Lambda\subset\mathbb{R}^{d}$ is a shifted lattice. In this case the
ellipsoid $E=\big{\\{}(\xi_{1},\ldots,\xi_{d})\ \ \bigg{|}\ \
\frac{\xi_{1}^{2}}{a^{2}}+\cdots+\frac{\xi_{d-1}^{2}}{a^{2}}+\frac{\xi_{d}^{2}}{a^{2}}+\frac{(\xi_{d+1}-\frac{1}{2})^{2}}{b^{2}}=1\big{\\}}\subset\mathbb{R}^{d+1}$
with $b>\frac{1}{2}$ and $a=\frac{2b}{\sqrt{4b^{2}-1}}$, is within the
$(b-\frac{1}{2})$-neighborhood of the region of $\mathbb{R}^{d+1}$ between the
hyperplanes $(\mathbb{R}^{d},0)$ and $(\mathbb{R}^{d},1)$ and satisfies the
following conditions: $E\cap(\mathbb{R}^{d},0)=(S^{d-1},0)$ and
$E\cap(\mathbb{R}^{d},1)=(S^{d-1},1)$. In particular, when
$\frac{1}{2}<b<\frac{3}{2}$ we have
$\operatorname{conv}(E)\cap(\Lambda\times\mathbb{Z})=\mathcal{E}\times\\{0,1\\}$.
∎

######  Remark 4.3.

The definition of a normal polytope in the introduction is stronger than the
one in [7, Definition 2.59]: the former is equivalent to the notion of an
_integrally closed_ polytope, whereas ‘normal’ in the sense of [7] is
equivalent to $\operatorname{gp}(P)$-normal. Examples of
$\operatorname{gp}(P)$-normal polytopes, which are not normal, are lattice
non-unimodular simplices, whose only lattice points are the vertices. Lemma
4.1 and the proof of Lemma 4.2 show that the 5-simplex $\Delta(5)$ is not
$\Lambda(6)$-unimodular. Applying an appropriate affine transformation we
obtain a lattice non-unimodular simplices
$\Delta^{\prime}\subset\mathbb{R}^{5}$ with
$\operatorname{vert}(\Delta^{\prime})$ ellipsoidal. Such examples in
$\mathbb{R}^{5}$ have been known sine the 1970s: a construction of Voronoi [2]
yields a lattice $\Lambda\subset\mathbb{R}^{5}$ and a 5-simplex
$\Delta\subset\mathbb{R}^{5}$ of $\Lambda$-multiplicity 2, whose circumscribed
sphere does not contain points of $\Lambda$ inside except
$\operatorname{vert}(\Delta)$.

We do not know whether there are ellipsoidal subsets
$\mathcal{E}\subset\mathbb{R}^{5}$ with $\operatorname{conv}(\mathcal{E})$
non-normal and
$\operatorname{gp}(\operatorname{conv}(\mathcal{E}))=\mathbb{Z}^{5}$. For
instance, $Q(5)$ is $\Lambda(5)$-normal, as checked by Normaliz [10].

Acknowledgment. We thank the referees for the streamlined version of the
original proof of Theorem (a), bringing [2] to our attention, and spotting
several inaccuracies.

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