Normal polytopes and ellipsoids

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 R^3 has a unimodular cover, and (3) for every d at least 5, there are ellipsoids in R^d, such that the convex hulls of the lattice points in these ellipsoids are not even normal. Part (3) answers a question of Bruns, Michalek, and the author.


Main result.
A convex polytope P ⊂ R d is normal if it is lattice, i.e., has vertices in Z d , and satisfies the condition ∀c ∈ N ∀x ∈ (cP ) ∩ Z d ∃x 1 , . . . , x c ∈ P ∩ Z d x 1 + · · · + x c = x.
A necessary condition for P to be normal is that the subgroup gp(P ) := x,y∈P ∩Z d Z(x − y) ⊂ 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 ≥ 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 ⊂ 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 ⊂ R 3 be a lattice polytope, E ⊂ R d an ellipsoid, and P (E) the convex hull of the lattice points in E.
(a) The unimodular simplices in P cover a neighborhood of the boundary ∂P in P if and only if P is very ample.
there exists E such that gp(P (E)) = Z d and P (E) is not normal. If in (c) we drop the condition gp(P (E)) = Z d , then ellipsoids E ⊂ R d with P (E) non-normal already exist for d = 5; see Remark 4.3.

1.2.
Preliminaries. Z + and R + denote the sets of non-negative integers and reals, respectively.
The convex hull of a set X ⊂ R d is denoted by conv(X). The relative interior of a convex set X ⊂ R d is denoted by int X. The boundary of X is denoted by ∂X = X \ int X.
Polytopes are assumed to be convex. For a polytope P ⊂ R d , its vertex set is denoted by vert(P ).
A lattice n-simplex ∆ = conv(x 0 , . . . , A unimodular pyramid over a lattice polytope Q is a lattice polytope P = 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 = R + x 1 + · · · + R + x k , where x 1 , . . . , x k ∈ Z d and C does not contain a nonzero linear subspace. For a cone C ⊂ R d , the smallest generating set of the additive submonoid C ∩ Z d ⊂ Z d consists of the indecomposable elements of this monoid. This is a finite set, called the Hilbert basis of C and denoted by Hilb(C). See [7, Chapter 2] for a detailed discussion on Hilbert bases. For a lattice polytope P ⊂ R d , we have the inclusion of finite subsets of Z d+1 : This inclusion is an equality if and only if P is normal.
A lattice polytope P is very ample if Hilb(R + (P − v)) ⊂ P − v for every vertex v ∈ 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 ⊂ R d , we say that C has a unimodular Hilbert triangulation (cover) if C can be triangulated (resp., covered) by cones of the form R + x 1 + · · · + Rx n , where {x 1 , . . . , x n } is a part of a basis of Z d as well as of Hilb(C).
An ellipsoid E ⊂ R d is a set of the form where l 1 , . . . , l d is a full-rank system of real linear forms and a 1 , . . . , a d ∈ R d . For a lattice polytope P , the union of unimodular simplices in P will be denoted by U(P ).

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.
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 ≥ 6 there are cones without unimodular Hilbert cover.
If P ⊂ R 3 is very ample, then by Theorem 2.1, for every v ∈ vert(P ), the cone R + (P − v) has a unimodular Hilbert triangulation: where T (v) is a finite index set, depending on v. In particular, the following unimodular simplices form a neighborhood of v in P : Also, lattice polygons have unimodular triangulation [7, Corollary 2.54]. Therefore, the following lemma completes the proof of Theorem (a): For a lattice polytope P of an arbitrary dimension, the following conditions are equivalent: (a) U(P ) is a neighborhood of ∂P within P ; (b) U(P ) is a neighborhood within P of every vertex of P and ∂P ⊂ U(P ).

Proof. The implication (a)=⇒(b) is obvious.
For the opposite implication, let: x ∈ ∂P ; F be the minimal face of P containing x; v ∈ vert(F ); T F be a unimodular cover of F with dim(F )-simplices, contained in F ; T v be a unimodular cover of a neighbourhood of v in P ; T v,F be the sub-family of T v , consisting of simplices that have a dim(F )dimensional intersection with F ; 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 conv(T v /F, T F ) covers a neighbourhood of x in P and consists of unimodular simplices.

Unimodular covers inside ellipsoids
3.1. Proof of Theorem (b). The set of normal polytopes P ⊂ R d carries a poset structure, where the order is generated by the elementary relation In [9] this poset is denoted by NPol(d). The trivial minimal elements of NPol(d) are the singletons from Z d . It is known that NPol(d) has nontrivial minimal elements for d ≥ 4 [7, Exercise 2.27] and the first maximal elements for d = 4, 5 were found in [9]. It is possible that NPol(d) has isolated elements for some d.
Computer searches so far have found neither maximal nor nontrivial minimal elements in NPol(3) [9]. The next lemma is yet another evidence that all normal 3-polytopes have unimodular cover.
Proof. If Q ≤ P is an elementary relation in NPol(d) and dim Q < dim P then P is a unimodular pyramid over Q. In this case every full-dimensional unimodular simplex ∆ ⊂ 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 ∈ NPol(3) has a unimodular cover if there is a 3-polytope Q ∈ NPol(3), such that Q has a unimodular cover and Q ≤ P is an elementary relation in NPol(3). Assume {v} = vert(P ) \ Q. By Theorem (a) we have the inclusion P \ U(P ) ⊂ Q. Since Q = U(Q) we have P = U(P ).
Call a subset E ⊂ Z d ellipsoidal and a point v ∈ E extremal if there is an ellipsoid E ⊂ R d , such that E = conv(E) ∩ Z d and v ∈ E.

Lemma 3.2. Let E ⊂ R d be an ellipsoidal set. Then E has an extremal point and E \ {v} is also ellipsoidal for every extremal point v ∈ E.
Proof. Let E = conv(E) ∩Z d for an ellipsoid E ⊂ R d . Applying an appropriate homothetic contraction, centered at the center of E, we can always achieve E ∩ E = ∅. In particular, E has an extremal point. For v ∈ E ∩ E, after changing E to its homothetic image with factor (1 + ε) and centered at v, where ε is a sufficiently small positive real number, we can further assume E ∩ E = {v}. Finally, applying a parallel translation to E by δ(z − v), where z is the center of E and δ > 0 is a sufficiently small real number, we achieve conv(E) ∩ Z d = E \ {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 E ⊂ Z d , there is a descending sequence of ellipsoidal sets of the form By [9, Theorem 6.5(c)], for d = 3, the conv(E i ) are normal polytopes. Therefore, * ≤ conv(E) in NPol(3) for some * ∈ Z 3 . Thus Lemma 3.1 applies.

3.2.
Alternative algorithmic proof in symmetric case. For the ellipsoids E with center in 1 2 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 Λ ⊂ R 2 and any ellipse E ′ ⊂ R 2 , such that conv(E ′ ) contains a triangle with vertices in Λ, every parallel translate conv(E ′ ) + v, where v ∈ R 2 , meets Λ.
Assume an ellipsoid E ⊂ R 3 has center in 1 2 Z 3 and dim(P (E)) = 3 (notation as in the theorem). Assume U(P (E)) P (E). Because ∂P (E) is triangulated by unimodular triangles, there is a unimodular triangle T ⊂ P (E), not necessarily in ∂P (E), and a point x ∈ int T , such that the points in [0, x], sufficiently close to x, are not in 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 ′ = conv(E) ∩ H is at least as large as the intersection of conv(E) with the affine hull of T : a consequence of the fact that P (E) ∩ Z 3 is symmetric relative to the center of E. The mentioned consequence of Johnson's theorem implies that conv(E ′ ) contains a point z ∈ Z 3 . In particular, all points in [x, 0], sufficiently close to x are in the unimodular simplex conv(T, z) ⊂ P (E), a contradiction.

High dimensional ellipsoids
For a lattice Λ ⊂ R d , define a Λ-polytope as a polytope P ⊂ R d with vert(P ) ⊂ Λ. Using Λ as the lattice of reference instead of Z d , one similarly defines Λ-normal polytopes and Λ-ellipsoidal sets.
only the case x ∈ 1 2 , . . . , 1 2 + Z d needs to be ruled out. Assume ξ i = 1 2 + a i for some integers a i , where i = 1, . . . , d. Then we have the inequalities Since the a i are integers we have d 4 ≥ d 2 − 1, a contradiction because d ≥ 5. One involves all dimensions d ≥ 6 by observing that (i) if E ⊂ R d is an ellipsoidal set then E × {0, 1} ⊂ R d+1 is also ellipsoidal and (ii) the normality of conv(E ×{0, 1}) implies that of conv(E). While (ii) is straightforward, for (i) one applies an appropriate affine transformation to achieve E = conv(S d−1 )∩Λ, where S d−1 ⊂ R d is the unit sphere, and Λ ⊂ R d is a shifted lattice. In this case the ellipsoid E = (ξ 1 , . . . , ξ d ) √ 4b 2 −1 , is within the (b − 1 2 )-neighborhood of the region of R d+1 between the hyperplanes (R d , 0) and (R d , 1) and satisfies the following conditions: E ∩ (R d , 0) = (S d−1 , 0) and E ∩ (R d , 1) = (S d−1 , 1). In particular, when 1 2 < b < 3 2 we have conv(E) ∩ (Λ × Z) = E × {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 gp(P )-normal. Examples of 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 ∆(5) is not Λ(6)-unimodular. Applying an appropriate affine transformation we obtain a lattice non-unimodular simplices ∆ ′ ⊂ R 5 with vert(∆ ′ ) ellipsoidal. Such examples in R 5 have been known sine the 1970s: a construction of Voronoi [2] yields a lattice Λ ⊂ R 5 and a 5-simplex ∆ ⊂ R 5 of Λ-multiplicity 2, whose circumscribed sphere does not contain points of Λ inside except vert(∆).