# Verification report — OWR-15587-006 (Michałek: lattice hulls of balls; very ample non-normal simplices)

Verification date: 2026-10-14.

**Verdict.** The note answers the first question in the negative (a counterexample, in every dimension n ≥ 6); the
second question was already answered in the literature, and the note claims nothing for it.
- **Settled: part (1) is false for n ≥ 6.** The lattice hull P of the closed ball in R⁶ with centre
  (0, 0, ½, 0, ½, 0) and squared radius 317813/2, i.e. P = conv{z ∈ Z⁶ : |z|² − z₃ − z₅ ≤ 158906}, is a
  full-dimensional spanning lattice polytope which is neither normal (in any convention) nor very ample (in either
  form) (Theorem A). For every n ≥ 6 the same inequality in n variables gives such a polytope (Corollary B). Three
  further certificates are given (Theorem 6.4), one in dimension 7. The proof is short; its only computer-assisted
  step is the enumeration of the lattice points of the ball on one hyperplane, done by three independent methods.
- **Part (2) is not new.** A very ample, non-normal lattice simplex is Example 2.3 of Cox, Haase, Hibi and
  Higashitani (2014), with very ampleness from their Theorem 1.1 and the remark on p. 2; Adeyemo, Bunnett and
  Levicán-Santibáñez (arXiv 2025) state this explicitly and thank W. Bruns. The source of the question cites the
  2014 paper as its reference [5]. The note gives a short proof for the reader's convenience (Proposition D).
- **Open.** Dimensions 4 and 5 (the face mechanism cannot work there: Proposition C); balls centred at lattice
  points, conv{z ∈ Zⁿ : |z|² ≤ N}; the smallest normal vector for which the construction works (10843 is only the
  smallest |n|² found by a non-exhaustive random search).
- **Novelty.** The counterexamples for round balls are new to our knowledge and relative to the literature found;
  for ellipsoids, non-normal examples in every dimension d ≥ 6 are due to Gubeladze (2021), with the same face
  mechanism. A search that finds nothing is not a proof of novelty; no priority is claimed.

Three independent AI-assisted verification runs (two on the first written version, one on the text of the note)
found no mathematical error. The note is unrefereed.

## Statement checked
- **Primary source.** M. Michałek, "Normal and Very Ample Polytopes - old and new open problems", Mini-Workshop
  "Lattice Polytopes: Methods, Advances, Applications", Oberwolfach Reports 14 (2017), no. 3, Report No. 44/2017,
  pp. 2684–2686, doi:10.4171/OWR/2017/44 (Crossref: report pp. 2659–2701, issued 2018). Read in the publisher's open
  PDF (content.ems.press, fetched 2026-10-14).
  - Setting (p. 2684): P ⊂ M ⊗ R = Rⁿ a spanning lattice polytope (0 ∈ P and Z(P ∩ M) = M); P is normal if for every
    k > 0 every point of kP ∩ M is a sum of k lattice points of P; very ample: the same "for k large enough".
  - Conjecture 1 (p. 2684): (1) "Let B be a ball in Rⁿ. Is conv(B ∩ Zⁿ) a normal polytope?", with the
    citation [4, Remark 6.6] ([4] = Bruns–Gubeladze–Michałek 2016); (2) does there exist a very ample, nonnormal
    simple polytope (or even a simplex)? Part (2) "would be a step towards" a counterexample to Oda's conjecture.
  - The reference list contains [5] = Cox, Haase, Hibi, Higashitani, EJC 21:4, P4.28 (2014), cited on p. 2684 for
    other questions.
- **Earlier source.** M. Michałek, "Toric structures in nontoric varieties", Oberwolfach Report 19/2016 (Toric
  Geometry), pp. 999–1001, doi:10.4171/OWR/2016/19: Conjecture 5(1) (p. 1000), the same question for "any ball",
  citing BGM Question 7.2(b) (ellipsoids); p. 1001: the first point "still awaits the answer". The whole report
  (pp. 963–1026) was read by the third verification run: it has no problem session; the question is one of two
  conjectures "presented during the lecture" in Michałek's abstract. Corpus record OWR-14601-005 consists of that
  abstract's Conjecture 3 and Conjecture 5(1), (2); its second question, "Is the convex hull of all lattice points in
  any ball a normal polytope?", is answered in the negative by the same counterexamples.
- **Corpus record.** ulamai/UnsolvedMath 1.6.0, OWR-15587-006 (data/problems.jsonl line 12385): statement = the
  source's two questions, with "(or even a simplex)" written out; upstream status `open`; category label "Group
  Theory" (it does not fit: the question is about lattice polytopes); its literature note (2026-08-21) says the
  simple-polytope and simplex versions remain unresolved, which overlooks the 2014 example.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (1) literally: every closed Euclidean ball, arbitrary centre and radius, every n | **no**: counterexamples for every n ≥ 6 | Theorem A, Corollary B |
| open balls instead of closed ones | same family of lattice hulls | Remark 2.3 |
| "normal" as IDP, as normality of the monoid in its group, or as in the source for spanning polytopes | the examples are spanning, so they fail all three | Definition 2.1, Lemma 2.2, Lemma 3.2(d) |
| "very ample" for all large k (source), or at every vertex (BGM Def. 5.2) | the examples fail both | Lemma 3.2(c) |
| per dimension | n ≤ 2 yes (Pick, as recalled in BGM §3.1); n = 3 yes (BGM Thm 6.5(c)); n = 4, 5 open; n ≥ 6 no | Section 1.3, Proposition C, Section 9 |
| balls centred at lattice points, conv{z : \|z\|² ≤ N} | open (not reached; observations only) | Section 9 |
| (2) existence of a very ample, non-normal simple polytope or simplex | **yes**, Cox–Haase–Hibi–Higashitani 2014 (known before the question was posed) | Proposition D, Section 8 |

## Results in the paper
- **Theorem A.** For n = (−36, −46, 50, −57, −41, 1): (a) n·z ≥ −41504 on the lattice points of P (Cauchy–Schwarz:
  2·10843·317813 = 6892092718 < 83019²); (b) the lattice points of P on n·z = −41504 are exactly six affinely
  independent points F₀..F₅, all on the boundary sphere, spanning a facet whose vertices generate a sublattice of
  index 2 (5×5 minors 72, −92, −100, −114, 82, 2); (c) w = ½ΣF_i = (413, 528, −572, 656, 472, −12) ∈ 3P is not a
  sum of three lattice points of P, and w + tF_j ∈ (3 + t)P is not a sum of 3 + t of them.
- **Lemma 3.2.** If the lattice points of P on a supporting hyperplane are affinely independent and a lattice
  point has non-integral barycentric coordinates with integer sum q, then P has no IDP, is not very ample in either
  form, and is not normal in any convention if spanning.
- **Corollary B.** Padding with zeros: conv{z ∈ Zⁿ : |z|² − z₃ − z₅ ≤ 158906} for every n ≥ 6 (uses that the six
  points lie on the sphere).
- **Lemma 6.1, Proposition 6.2.** A ball touching the hyperplane n^⊥ from one side turns an empty-sphere simplex of
  n^⊥ ∩ Z^d into a face of its lattice hull. **Lemma 6.3** moves the centre to a prescribed point when an integrality
  and parity condition holds (half-integral centres of Theorem A and Theorem 6.4(b); no lattice-point centre for
  these simplices).
- **Theorem 6.4.** Three further certificates: a ball with rational centre for n = (387, 25, −279, −355, 219, 1);
  conv{z ∈ Z⁶ : |z|² − z₂ − z₅ ≤ 25598296}; a ball in R⁷ with an empty 6-simplex facet of index 2.
- **Proposition C.** A face of the lattice hull of an ellipsoid that is an empty simplex of dimension ≤ 4 is
  unimodular (box-point argument with BGM Theorem 6.5(b)); so the mechanism needs dimension ≥ 6.
- **Proposition D** (Cox–Haase–Hibi–Higashitani). For m ≥ 4, Q = 3 conv(0, e₁, …, e_{2m−2}, (m−1, …, m−1, m)) is
  a spanning, very ample, non-normal simplex; for m = 4 it is 3 conv(0, e₁, …, e₆, (3, 3, 3, 3, 3, 3, 4)) ⊂ R⁷.

## Computations (programs and outputs in reproducibility/)
Only the enumerations of Theorem A(b) and of the proof of Theorem 6.4 are computer-assisted; every other theorem,
proposition, corollary and lemma is proved in the text. The paragraph after the proof of Proposition D, Section 9 and
Table 2 report computer observations that no proof uses.
- **The program of the note** (`check_note.py`, standard library, about 2 s, 79 checks, ALL CHECKS PASSED): Theorem A
  with three enumerations of the slice (3125 tuples with z₆ solved; the full box of 15625 points; Fincke–Pohst with
  exact LDLᵀ), the bound, the minors and index, the witness, spanning, two controls (bound 158907: 14 points on the
  slice; centre 0: the bound fails); the padded balls in Z⁷, Z⁸; the three certificates of Theorem 6.4; and
  Proposition D for m = 4 (box points, the 128 lattice points of Q, the indecomposable point (3,…,3) of 2Q,
  spanning, all 15784 lattice points of 3Q decompose).
- **Programs with which the results were first obtained** (`original/`): `verify_ball.py` (the hypotheses of
  Proposition 6.2; complete enumerations of 161051, 59049 and 1771561 candidate tuples for Theorem 6.4(a), the
  section of Theorem A and Theorem 6.4(c)), `verify_direct.py` (Theorem A and Theorem 6.4(b) from the balls; 59049
  and 161051 tuples), `independent_check_thmA.py` (full box of 531441 points), `verify_padded.py` (d = 7, 8),
  `verify_centered.py` (Lemma 6.3), `verify_simplex_chhh.py` (Proposition D, m = 4), and the search programs.
- **The search** (`search_sections.py`): sections n = (a, 1), a random in [−A, A]⁵ with non-zero entries of distinct
  absolute values; exact LLL, Fincke–Pohst, floating-point Delaunay triangulation (scipy) to find candidates, exact
  empty-sphere check. 4320 sections sampled (bounds A = 20, 35, 60: 980 sections, none with simplices; A = 80, 100,
  150, and 200, 400 with odd entries), 17 sections with index-2 empty-sphere simplices, 204 records (12 per
  section). The smallest |n|² among them, 10843, is the section of Theorem A — only the smallest found by a
  non-exhaustive random search. In all 17 sections the centres admitted by Lemma 6.3 contain no lattice point
  (`analyze_hits.py`; confirmed by the third verification run); two exploratory searches for lattice-point centres (`search_origin.py`, stopped early;
  `search_origin2.py`, all 10843 slice classes for the section of Theorem A) found none. These are observations.
- **Re-runs.** On 2026-10-14 all 23 programs of the package that have a recorded output and need neither a
  third-party PDF nor numpy/scipy were run again from a copy (`rerun_quick.py`); all outputs are identical to the
  recorded ones up to running times (`reproducibility/RERUN_LOG.txt`). The third verification run repeated this from
  an extracted copy of the source archive: 23 of 23 identical (`reproducibility/independent_run_2/package_rerun.log`).
- **The third verification run** (`reproducibility/independent_run_2/`, own exact library, written from the text of
  the note): self-test of its level enumerator against naive enumeration (300 random slices, 0 mismatches); Theorem
  A with three further enumerations of the facet level (Fincke–Pohst in an LLL-reduced basis; z₁ solved from the
  hyperplane; a full box with exact per-axis extents), Remark 4.1, Corollary B for n = 7, 8, 9 and the Lemma 6.3
  numbers (47 checks); Theorem 6.4 and the Lemma 6.3 numbers for the data (a) (45 checks); Proposition D for m = 4
  (counts from the inequality description, box points, decompositions up to 4Q₄, corner cones; 12 checks) and m = 5
  (up to 4Q₅; 10 checks); the 204 search records, Table 2 and the lattice-point-centre observation (9 checks). All
  pass.

## Independent verification runs
The results were first obtained, with proofs and the programs in `original/`. Two independent verification runs,
both AI-assisted, followed; each wrote its own programs without using those of the first version. Run A examined
the statement, the proofs and the sources; run B recomputed everything with its own exact library and examined the
literature and novelty. A third run examined the written note (see "Third verification run" below).

| Item | Run A | Run B |
|---|---|---|
| Statement, definitions and readings against the source | CONFIRMED (source read; readings R0–R4 discussed; the source cites CHHH 2014 as [5]) | CONFIRMED (source read; same observation on [5]) |
| Lemma 3.1, Lemma 3.2 (first version: Lemma 2, Proposition 3) | CONFIRMED (re-derived, including the non-integrality step and the corner-cone form) | CONFIRMED (re-derived) |
| Theorem A | CONFIRMED (re-derived; two enumerations of its own and a third, general program; 31 + 85 checks) | CONFIRMED (three enumerations of its own; validated against naive enumeration on 400 random balls and 11 wrong inputs) |
| Corollary B | CONFIRMED_WITH_FIXES (state it for the ball of Theorem A; padding recomputed for d = 7, 8) | CONFIRMED (d = 7, 8, 9 recomputed) |
| Lemma 6.1, Proposition 6.2, Lemma 6.3 | CONFIRMED (re-derived; the numbers 91, ω, τ, R² recomputed) | CONFIRMED (parity data recomputed) |
| Theorem 6.4 (a), (b), (c) | CONFIRMED (recomputed from the balls) | CONFIRMED (recomputed from the balls) |
| Proposition C | CONFIRMED (re-derived, including the proof of BGM Theorem 6.5(a),(b)) | CONFIRMED |
| Proposition D (part (2)) | CONFIRMED_WITH_FIXES (citation of CHHH: m ⩾ 4; very ampleness via Theorem 1.1 and the p. 2 remark); re-derived for all m ≥ 4; computation for m = 4 | CONFIRMED (computations for m = 4 and m = 5 with an exact corner-cone certificate) |
| The 204 search records | not re-checked | CONFIRMED (all 204 by its own pipeline; 17 sections; smallest found 10843) |
| Programs of `original/` | not read; their recorded outputs agree with the run's own numbers | re-run from a copy: 10 of 10 outputs byte-identical |
| Literature and novelty | no publication on round balls found; Gubeladze 2021 closest (ellipsoids) | the same; credit for (2) to CHHH 2014 and ABL 2025 |

No run found a wrong theorem, proposition or lemma, or a gap.

**Corrections required by the runs**, all applied in the note:
1. (A) CHHH state Example 2.3 for m ⩾ 4 (text extraction shows ">"): Section 8 and Proposition D.
2. (A) CHHH show that 3P is not IDP and that μ_Hilb(P) = 2; very ampleness of 3P follows from their Theorem 1.1
   (μ_va ≤ μ_Hilb) and the remark on p. 2: Sections 1.2 and 8.
3. (A) The exact content of ABL 2025 (thanks to Bruns for informing them about a very ample non-normal
   non-rectangular simplex; Section 2.3 cites CHHH Example 2.3 for a 7-dimensional one): Section 1.2, abstract,
   "Scope and priority".
4. (A) The source cites CHHH 2014 as its reference [5]: Section 1.2.
5. (A) The statements on Delaunay simplices of lattices of rank ≤ 4 and 5 attributed to Voronoi/Baranovskii were
   not checked against a source: removed.
6. (A) The statement that the search re-found the simplex of Theorem 6.4(a) has no recorded output: removed; the
   note says that (a) and (c) come from an exploratory search whose program is not included.
7. (A) n ≤ 3: n = 3 is BGM Theorem 6.5(c); n ≤ 2 by Pick's theorem: abstract, Section 1.3.
8. (A) Semantic Scholar returned no citations even for BGM 2016: its empty result is reported as carrying no
   information ("Scope and priority").
9. (A) Corollary B is stated and proved for the ball of Theorem A (all F_i on the sphere).
10. (A) The scope (arbitrary centres; n = 4, 5 and lattice-point centres open; novelty only as far as searched) is
    stated in the abstract, Section 1 and "Scope and priority".
11. (B) The smallest |n|² found (10843) is described as only the smallest found by a non-exhaustive random
    search: Sections 9 and 10.
12. (B) Bibliography: Gubeladze 2021 = EJC 28(4) P4.7; Gubeladze's survey = JPAA 227(2) (2023) 107187; CHHH DOI
    10.37236/4204 (all checked with Crossref).

**Changes made when the note was written** (not seen by the runs): `check_note.py`; the last sentence of
Lemma 6.3(b) (for lattice-point centres the condition does not depend on the centre); the remark on the spanning
condition after Proposition 6.2; Remark 4.1 (controls); the remark that projective toric varieties of the examples
are not normal (from the dictionary recalled in the source); the remark that Gubeladze's lattice Λ(d) is not
similar to Z^d (minimal norm 1, determinant ½), so his ellipsoids are not round balls; the numbering of the
statements.

### Third verification run (on the written note)
After the note was written, a third verification run, also AI-assisted, examined its text line by line as if
nothing had been checked before (the runs A and B had examined the first written version). It read the sources
again (the report 44/2017; the whole report 19/2016; BGM 2016, arXiv v2; Gubeladze 2021, journal version and arXiv
v2; CHHH 2014, journal version; ABL 2025, v1, still the only version; Bruns 2013, Section 8; Gubeladze's survey,
arXiv v2), re-derived every proof of the note, including the proofs of BGM Theorem 6.5(a),(b) and of μ_va ≤ μ_Hilb
in CHHH Theorem 1.1, wrote its own exact programs (`reproducibility/independent_run_2/`, results above), re-ran the
package from the source archive (23 of 23 identical) and repeated the literature search (arXiv API, Crossref,
OpenAlex, zbMATH Open, one web search; requests in `independent_run_2/requests.log`).

| Item | Third run |
|---|---|
| Statement, conventions, class of balls, 2016 occurrence, corpus record | CONFIRMED (the 2016 version is in Michałek's abstract, p. 1000, not in a problem session; it is part of record OWR-14601-005) |
| Lemmas 3.1, 3.2, 6.1, 6.3; Proposition 6.2; Theorem A; Corollary B; Theorem 6.4; Propositions C and D | CONFIRMED (re-derived; all finite facts recomputed) |
| Cited results (statements, hypotheses, numbering) | CONFIRMED |
| Search records, Table 2, Section 9 observations | CONFIRMED (the two exploratory searches for lattice-point centres were not repeated) |
| Novelty | no publication on lattice hulls of round balls found; Gubeladze 2021 remains the closest work |

It found no mathematical error and no gap. **Its required corrections, all applied in the note:**
1. The 2016 occurrence of question (1) described precisely (Michałek's abstract in OWR 19/2016, Conjecture 5(1),
   p. 1000, quoted; p. 1001), with the remark that it is part of record OWR-14601-005 (Section 1.1).
2. "who credit W. Bruns" replaced by "who thank W. Bruns for informing them of it" (abstract, "Scope and priority").
3. Credit to Voronoi's construction of empty-sphere 5-simplices of multiplicity 2, as recalled in Gubeladze 2021,
   Remark 7 (Section 1.3).
4. The reflection b ↦ b* in the proof of Proposition C credited to BGM Lemma 2.3.
5. "[Gub21, after Remark 7]" corrected to "[Gub21, Remark 7, last paragraph]" (Section 9).
6. Gubeladze's condition gp(P(E)) = Z^d described as "the lattice points affinely generate Z^d" (Sections 1.3, 7, 9).
7. The first sentence of Section 10 restricted to the numbered results; the computer observations are named.
8. The padded polytope of Corollary B renamed P^(n) (it clashed with the simplex P_m of Proposition D).
9. This run recorded in the note ("Verification", "What was read", searches) and in this report.
10. Its programs and outputs added to the package (`reproducibility/independent_run_2/`).

## Relation to the literature, novelty and scope
- **Searches (2026-10-14; when the results were first obtained, by all three runs, and at the writing).** arXiv API
  keyword queries (ball/sphere/ellipsoid/lattice points/empty sphere/Delaunay × normal polytope/integer
  decomposition property/integrally closed/very ample); Crossref for all DOIs; OpenAlex citing works (9 for BGM
  2016, among them Gubeladze 2021; none for Gubeladze 2021; none for the report 19/2016; 3 for Gubeladze's survey;
  3 items for the report containing the source; 32 for CHHH 2014, none on balls); zbMATH Open keyword queries;
  Semantic Scholar (no coverage); seven web searches in all.
  - No publication answering question (1) for round balls was found.
  - Closest: Gubeladze, Normal polytopes and ellipsoids, EJC 28(4) (2021) P4.7, Theorem (c): non-normal lattice
    hulls of ellipsoids generating Z^d for every d ≥ 6, and for d = 5 without the generation condition (Remark 7);
    the ellipsoids are linear images of the ball circumscribing [0,1]^d for the lattice Z^d + Z(½,…,½), not round
    balls.
  - BGM 2016: Theorem 6.5(b) (degree 2 for all ellipsoids), 6.5(c) (n = 3), Remark 6.6(a) (tests in dimensions 4
    and 5), Question 7.2(b) (ellipsoids).
  - Part (2): CHHH 2014 Example 2.3 (with Theorem 1.1 and the p. 2 remark); ABL 2025 explicit; Bruns 2013, Section 8
    (origin of the question).
- **What was read.** The source (pp. 2684–2686) and OWR 19/2016 (pp. 999–1001) in the publisher's PDFs; BGM 2016 in
  arXiv v2 (numbering used; journal version not compared); Gubeladze 2021 in the journal version (numbering used)
  and arXiv v2; CHHH 2014 in the journal version (introduction, Theorem 1.1 with proof, Examples 2.2–2.5); ABL 2025
  v1 (introduction, acknowledgements, Section 2.3); Bruns 2013, Section 8; Gubeladze's survey, arXiv v2 (searched
  for balls and ellipsoids). The third run read these again, and the whole report 19/2016. Proofs of cited theorems
  were not checked, except BGM Theorem 6.5(a),(b) and the inequality μ_va ≤ μ_Hilb of CHHH Theorem 1.1 (re-derived by
  run A and by the third run). All DOIs were checked with Crossref (again by the third run).
- **Caveats.** Michałek's "normal" is defined for spanning polytopes; the examples are spanning, so no convention
  issue arises. Under the narrower reading "balls centred at lattice points" the question stays open. This
  negative search is not a proof of priority.
- **Scope.** Question (1): answered negatively in every dimension n ≥ 6, for balls with arbitrary centres.
  Question (2): known (CHHH 2014); no claim.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/