# Verification report — OWR-15215-006 (Meckes's monotonicity question for random simplices, recorded by Reitzner, Oberwolfach Report 10/2017, p. 519)

Verification date: 2026-10-15.

**Verdict.** The note answers the second question of the record (a lower bound for the optimal constant) on
the exponential scale, and gives a self-contained explicit version of the known answer to the first question.
Every statement it makes is proved, or quoted from an exactly cited published theorem.
- **Settled in print (not new).** Question 1 (does c(d) with lim c(d)^(1/d) < ∞ exist?) has a positive answer:
  Rademacher, Mathematika 58 (2012), Appendix A, Proposition A.4 (Proposition 20 of arXiv:1008.3944v1:
  Meckes's weak conjecture, for every moment p ≥ 1, is equivalent to the slicing conjecture), together with
  Klartag–Lehec, GAFA 35 (2025), Theorem 1.2 (sup_n L_n < ∞). Reichenwallner–Reitzner, Mathematika 62 (2016),
  Section 1, and the source itself also state the equivalence.
- **Self-contained explicit version (Theorem 1.1).** c*(d) ≤ θ_d (L_d/L_{B^d})^d with
  θ_d ≤ ((d+1) C(d+8,8))^(1/2), hence limsup c*(d)^(1/d) ≤ √(2πe) sup_n L_n; it uses Rademacher's identity
  E V^2 = (d+1)/d! det Cov (Lemma 2.1, proved), Groemer's theorem (cited) and Klartag–Lehec (cited).
- **New, to our knowledge.** Theorem 1.2 and Corollary 1.3: liminf c*(d)^(1/d) ≥
  (300657950289351/292457950289351)^(1/2) = 1.0139221956…, so c*(d) = e^{Θ(d)}; explicitly
  c*(d) ≥ [(d+1) ∏_{r≤d}(1 + 26.8/r)]^(-1/2) · 1.28257669^(⌊d/9⌋/2) ≥ [(d+1) C(d+27,27)]^(-1/2) ·
  1.28257669^(⌊d/9⌋/2) for every d. The middle bound exceeds 1 if and only if d ≥ 6525, the right-hand one if
  and only if d ≥ 6570; **for d ≤ 6524 the explicit bound is weaker than the trivial c*(d) ≥ 1**, and the
  asymptotic statement is the main result. The product bound behind it (Theorem 6.1) and the explicit pair
  (Proposition 5.3) are, as far as the searches went, not in print; the product idea may be folklore.
- **Open.** The exact value of c*(d) for d ≥ 3; the existence and value of lim c*(d)^(1/d); explicit
  non-trivial lower bounds for 3 ≤ d ≤ 6524 (a Monte Carlo experiment, evidence only, suggests c*(9) ≥ 1.1).

The note is unrefereed.

## Statement checked
- **Primary source.** M. Reitzner, *High-dimensional random polytopes*, extended abstract in Oberwolfach
  Report 10/2017 (Mini-Workshop: Perspectives in High-Dimensional Probability and Convexity, 5–11 February
  2017, organisers J. Prochno, C. Thäle, E. M. Werner), Oberwolfach Reports 14(1), pp. 499–526,
  doi:10.4171/OWR/2017/10; the abstract is on pp. 518–520, both questions on p. 519.
  - Read in the publisher's open PDF (ems.press/content/serial-article-files/46672; 294,552 bytes, sha256
    `7a47e581bd272137291961e83e03851ad99940d081f4f62b0bf20e5edb5dffcd`), downloaded on 2026-10-15.
  - Question 1 (attributed to Meckes): does there exist c(d) with lim_{d→∞} c(d)^(1/d) < ∞ such that
    E V_d(P^K_{d+1}) ≤ c(d) E V_d(P^L_{d+1}) for all K ⊂ L? The text adds that this is equivalent to a positive
    solution of the slicing problem (no proof there).
  - Question 2 ("Open Problem"): c(d) is defined as a minimum over K ⊂ L of the ratio of the two expectations,
    and the reader is asked to compute a lower bound for c(d), after the remark that it would be of interest to
    compute the lower bound for c(d) given by Rademacher's counterexamples.
- **Origin.** M. Meckes, problem 4, *Monotonicity of volumes of random simplices*, in the problem list of the
  AMS Special Session "Recent Trends in Convex and Discrete Geometry", San Antonio, January 13–14, 2006
  (math.gmu.edu/~vsoltan/SanAntonio_06.pdf, read): conjecture M(K_1) ≤ M(K_2) for K_1 ⊂ K_2, and the "Weaker
  Conjecture" M(K_1) ≤ a^n M(K_2) with a universal a, with M(K) the mean volume of a random simplex in K.
- **Corpus record.** ulamai/UnsolvedMath, OWR-15215-006 (dataset version 1.6.0; upstream status
  `partially_solved`). Its statement writes limsup and ends "Equivalently, compute a lower bound for the
  optimal such c(d)."

## Readings
| Reading | Answer | Where |
|---|---|---|
| Q1 with "lim" read as limsup (as the record writes) | yes, c(d) = a^d (in print); explicit form in Theorem 1.1 | §1.2, Theorem 1.1, §4 |
| Q1 with "lim" read literally | the same answer satisfies it (the limit of a^(d/d) exists) | §1.1 |
| Q2 literally: inf over K ⊂ L of E V_K / E V_L | degenerate: equals 0 (homothetic shrinking), not attained | §1.1 |
| Q2 as a lower bound for the optimal constant c*(d) = sup E V_K / E V_L (the record's reading) | exponential lower bound, Theorem 1.2, Corollary 1.3 | §1.1, §6 |
| Q2 as min of E V_L / E V_K = 1/c*(d) | a lower bound for it is an upper bound for c*(d), i.e. Q1 again (Theorem 1.1) | §1.1 |

## Results in the paper
- **Lemma 2.1** (Rademacher's Lemma 4, proof included): E V^2 = (d+1)/d! det Cov for i.i.d. points.
- **Theorem 2.2** (Groemer, cited) and the ball formulas of Miles and Kingman (cited; used only for numbers).
- **Lemma 3.2** (Gram–Schmidt and Hölder): under the fourth-moment condition (H) (Definition 3.1),
  E V ≥ [(d+1) ∏_{r≤d}(1 + C_0/r)]^(-1/2) (E V^2)^(1/2), and ∏(1 + C_0/r) ≤ C(d+N, N) for an integer N ≥ C_0.
- **Lemma 3.3** (ball: (H) with C_0 = 8). **Lemma 3.4** (products: (H) with C_0 = max(8, 4 + 2 max Λ_a),
  independent of the number of blocks; Λ_a ≤ ρ^2/σ^2 if |Z^a| ≤ ρ). **Remark 3.5** (cube:
  log E V_{[0,1]^d} = −½ log d! − (d/2) log 12 + O(log d), with explicit two-sided bounds; no novelty claimed).
- **Theorem 1.1** (proved in §4) and Remarks 4.1 (θ_1 = 1.2247, θ_10 = 1.6712, θ_1000 = 4.9072;
  (θ_d/L_{B^d}^d)^(1/d) ∈ [3.99, 4.25] for d ≤ 1000, floating point) and 4.2 (comparison with Rademacher's
  route; conditional bound √(2π/e) = 1.5203 if sup L_n = 1/e).
- **Lemma 5.1**: for 0 < h ≤ (n+1)/(n−1) (β ≥ 1), L_0 = Δ ∪ pyramids = {λ_j + βλ_i ≥ 0} is a convex polytope
  and K_0 = Δ ∪ (pyramids cut at the fraction s_0 of their height) is convex. Sentence after it: non-convex for
  larger h; at h = (n+1)/(n−1) adjacent facets are coplanar, so L_0 has n(n+1)/2 facets (45 for n = 9).
- **Lemma 5.2**: closed form of the scalar covariances σ^2(t) (regular simplex, inradius 1; symmetry and
  irreducibility of the standard representation of S_{n+1}).
- **Proposition 5.3**: n = 9, h = 5/4, s_0 = 1/11: σ_L^2 = 149/176, σ_K^2 = 300657950289351/345453686247824,
  ratio 300657950289351/292457950289351, R = det A(K_0)/det A(L_0) = ratio^9 = 1.2825766950… > 1; K_0 keeps
  of each pyramid only the bottom 1/11 of its height (57.6% of its volume); vol K_0 = 1.7199 vol Δ,
  vol L_0 = 2.25 vol Δ. Pairs for n = 3, 4, 5 with ratios 1.013123, 1.048589, 1.094243.
- **Lemma 5.4**: E|Z|^4 = 675609637612497091767/8694378375485234432, E[|Z|^2 ZZ^T] scalar,
  Λ_0 = 11.3985358…, and the crude Λ_0 ≤ q_0 = 81/σ_K^2 = 93.068.
- **Theorem 6.1** (products K_0^m × [−1,1]^j ⊆ L_0^m × [−1,1]^j): c*(d) ≥ [(d+1) ∏(1 + C_0/r)]^(-1/2) R_0^(m/2),
  liminf c*(d)^(1/d) ≥ R_0^(1/(2 d_0)).
- **Lemma 6.2** (thresholds: B(d) decreases inside each block of 9; the positive set is an up-set in each
  residue class; two exact comparisons decide the threshold), **Theorem 1.2** (proof in §6),
  **Corollary 1.3**, **Remark 6.3** (exact constants: threshold 6525; crude C_0 = 190.137: threshold 44694;
  d-th roots 0.9456, 1.003318, 1.012497 at d = 900, 9000, 90000), **Remark 6.4** (Vempala's constant c_V:
  supermultiplicative, lim c_V(d)^(1/d) = γ_V ≥ R^(1/9) = 1.0280382, liminf c*(d)^(1/d) ≥ γ_V^(1/2)),
  **Remark 6.5** (Monte Carlo, evidence only: E V_{K_0}/E V_{L_0} = 1.109 ± 0.002, 1.108 ± 0.002,
  1.110 ± 0.001 and 1.111 ± 0.002 in four independent samplers).
- **§7** open questions: the exact rate; whether sup θ(L) = e^{o(d)} (then lim c*(d)^(1/d) = γ_V^(1/2));
  better pairs; rigorous values for small d.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation beyond the exact evaluation of stated rational numbers and integer
inequalities. `reproducibility/check_note.py` (Python ≥ 3.8, standard library only, exact rational arithmetic in
parts (i)–(v), fixed seed; 19.4 s; 51 checks, TOTAL failures: 0):
- (i) the n = 9 pair by an exact decomposition into simplices (Δ + 10 pyramids; Δ + 90 staircase simplices,
  cross-checked by the signed decomposition), Dirichlet moments up to order 4: volumes, centroids, scalar
  covariances (exact test), σ_L^2 = 149/176, σ_K^2, the ratio, the determinant ratio in affine coordinates without
  symmetry (= ratio^9 exactly), the closed form of Lemma 5.2, E|Z|^4, scalar E[|Z|^2 ZZ^T], Λ_0, C_0, q_0;
- (ii) the same for n = 3, 4, 5;
- (iii) the thresholds of the four explicit bounds by exact integer comparisons and Lemma 6.2: 6525 (exact
  constants), 6525 (26.8 and 1.28257669), 6570 (binomial form), 44694 (crude constant); R^(1/18) to 15 digits;
- (iv) negative controls: s_0 = 1/10 changes σ_K^2, the ratio and the threshold (6552); a 1% apex perturbation
  destroys the scalar covariance; for h = 3/2 the midpoint of two apexes is outside the union;
- (v) Lemma 5.1: inequalities at the vertices, 45 facets, 1,500 random rational points (H- versus
  V-description, no disagreement);
- (vi) floating point: Miles (k = 1, 2) versus Kingman and Lemma 2.1 (log difference ≤ 1.3e-9), the
  classical values for d = 1, 2, 3, θ_d ≤ ((d+1) C(d+8,8))^(1/2), and the range [3.99203, 4.24264].

The package also contains the programs with which the results were first obtained (`original/`, nine programs:
exact closed forms, exact triangulation with certified supporting hyperplanes for n = 3, 4, 5, floating-point hull
integration for n = 9, exact fourth moment, Monte Carlo, explorations) and those of the three verification runs
(`verification_run_A/`, `verification_run_B/`, `independent_run_2/`).

## Independent verification runs
The results were first obtained, with proofs and nine programs, in a first written version. Two independent
verification runs, both AI-assisted, examined that version (2026-10-15); each re-derived the proofs and wrote its
own programs. The note was then written with all corrections; its program `check_note.py` tests the final
statements. A third independent verification run, also AI-assisted, then examined the final text of the note and
the package (2026-10-15); it is described after the corrections of the first two runs.

| Item | Run A | Run B | Writing stage |
|---|---|---|---|
| Statement against the source (both questions, p. 519; Meckes 2006) | CONFIRMED | CONFIRMED | CONFIRMED (publisher's PDF and Meckes's list read again) |
| Reading of Q2 as the optimal constant | CORRECT (three readings examined) | agreed | stated in abstract and §1.1 |
| Q1 in print (Rademacher + Klartag–Lehec) | CONFIRMED (arXiv numbering caveat) | CONFIRMED (published Appendix inferred) | published text read (author's web copy): Proposition A.4 of Appendix A |
| Lemma 2.1, Lemma G (3.2), Lemma B (3.3), Lemma P (3.4) | CONFIRMED; tested on exact discrete laws | CONFIRMED; (H) attacked numerically (max ratio 0.27) | CONFIRMED |
| Theorem 1.1 (explicit upper bound) | CONFIRMED | CONFIRMED (Miles vs Lemma 2.1 vs Kingman, d ≤ 1000) | CONFIRMED; check_note (vi) |
| Lemmas S1, S2 (5.1, 5.2); the pair (5.3) | CONFIRMED; exact recomputation without symmetry; 50,000 exact random points; hull volumes | CONFIRMED; signed decomposition, four parameter sets; wrong inputs detected; facet structure | CONFIRMED; check_note (i), (ii), (v) |
| Fourth moment, Λ_0, C_0 | CONFIRMED (exact) | CONFIRMED (exact) | CONFIRMED (exact, scalar matrix) |
| Theorem 2 (6.1), Corollary 3 (Theorem 1.2), thresholds | CONFIRMED; 6525 / 6570 / 44694 | CONFIRMED; 6525 / 6570 / 44694 | CONFIRMED; "if and only if" proved by Lemma 6.2 and exact integer comparisons |
| Vempala constant remark (6.4) | CONFIRMED | CONFIRMED | CONFIRMED |
| Monte Carlo (evidence only) | 1.10822 ± 0.00210 | 1.1102 ± 0.0014 | quoted as evidence (Remark 6.5) |
| Literature, novelty | no prior quantitative bound found | no prior quantitative bound found | searches repeated; none found |

No run found a false theorem, proposition or lemma, or a gap in a proof.

**Corrections required by run A**, all applied in the note:
1. The explicit bound exceeds the trivial value 1 only for d ≥ 6525 (6570 for the binomial form): stated in the
   abstract, in §1.3 after Theorem 1.2, in Theorem 1.2 itself ("for d ≤ 6524 the bound is weaker than the
   trivial bound c*(d) ≥ 1"), in §8 and in "Scope and priority".
2. Numbering of Rademacher's proposition: the note cites Appendix A, Proposition A.4 of the published version
   and gives Proposition 20 of arXiv:1008.3944v1 (§1.2, bibliography); the published text was read in the
   copy on the author's web page (see the literature section below).
3. Reichenwallner, Monatsh. Math. 186 (2018), no. 3, 481–501, published online 21 June 2017 (bibliography).
4. The reading of Q2 is supported by the corpus record's own statement ("a lower bound for the optimal such
   c(d)"), quoted in §1.1; a wording taken from another copy of the record is not used.
5. The threshold of each form is stated: 6525 for the product form with 26.8 (and with the exact constants),
   6570 for the binomial form, 44694 for the crude constant (Theorem 1.2, Remark 6.3).
6. K_0 is described precisely: of each pyramid (height 5/4 in units of the inradius) only the slab between the
   base and the parallel hyperplane at 1/11 of the height is kept (57.6% of the pyramid's volume);
   vol K_0 = 1.7199 vol Δ, vol L_0 = 2.25 vol Δ (abstract, §1.3, §5, Proposition 5.3).

**Recommendations of run B**, all followed: Kingman (1969), J. Appl. Probab. 6, 660–672, cited for the closed
form (§2, with a note that only its abstract was read); the coplanar facets of L_0 at the certified parameters
(45 facets, not 90; sentence after Lemma 5.1, checked by check_note (v)); the status of Rademacher's Appendix
and arXiv:1008.3944 alongside the journal version (resolved by reading the published text, see below); the
year of Reichenwallner's paper (2018, online 2017).

**Changes made when the note was written, after runs A and B.**
- The published version of Rademacher's paper was found on the author's web page and read; there the
  equivalence is Proposition A.4 of Appendix A (Lemma A.3 is the Khinchine-type inequality, Theorem A.2
  Klartag's theorem); Lemma 4 and Theorems 1, 3, 6, 7 have the same numbers in both versions.
- Lemma 6.2 was added: it turns "the least d with bound > 1" into "bound > 1 if and only if d ≥ d*", certified by
  two exact integer comparisons per form (check_note.py (iii)).
- The cube remark (Remark 3.5) is stated with its two-sided bound and proof; no novelty is claimed.
- The open question on θ(L) and the relation of (H) to the variance conjecture (§7) were added.

**Third independent verification run (examination of the final text).** It worked on the text of the note and
the source archive as if nothing had been checked before, with its own programs (`reproducibility/independent_run_2/`,
with README and recorded outputs), and found no mathematical error.

| Item | What the third run did | Result |
|---|---|---|
| Statement against the source | read the publisher's PDF again (p. 519 also inspected as an image) and Meckes's list; compared both questions, the displayed definition of c(d) and the record's wording | CONFIRMED; the record's "Equivalently" merges two separate requests of the source (a matter for the corpus record, not for the note) |
| Cited theorems | read Rademacher's published version (author's web copy) and arXiv v1, Klartag–Lehec arXiv v1 (Section 1), Reichenwallner–Reitzner arXiv v1, Kunis–Reichenwallner–Reitzner arXiv v1, Reichenwallner's published open version; Crossref for all 11 DOIs | CONFIRMED, except the numbering sentence and the page locator corrected below |
| Proofs | re-derived Lemmas 2.1, 3.2, 3.3, 3.4, 5.1, 5.2, 5.4, 6.2, Theorems 1.1, 1.2, 6.1, Corollary 1.3, Remarks 3.5, 6.4 | CONFIRMED; Remark 3.5 (cube) is fully proved |
| The pair (Proposition 5.3, Lemma 5.4, n = 3, 4, 5) | own exact program: polytopes from their inequality descriptions, pulling triangulation with certified closed boundary, exact polarization formula for polynomial integrals over simplices (tested against symbolic integration); no pyramid decomposition, no symmetry argument | all rational values reproduced exactly; controls detected |
| Thresholds | exact integer comparisons for the hypotheses of Lemma 6.2(c), 40-digit scans | 6525, 6525, 6570, 44694; 6552 for s_0 = 1/10 |
| Ball constants (Remark 4.1) | mpmath, 50 digits | all values reproduced |
| Lemmas 3.2, 3.4 | exact tests on a non-symmetric discrete law with scalar covariance | no violation (max ratio 0.398) |
| Monte Carlo (evidence only) | own sampler on its triangulation, 2·10^6 simplices per body | 1.1111 ± 0.0021 |
| Package | every command line of the package README re-run from an unpacked copy of the archive | every recorded output reproduced up to running times and output format; two recorded outputs of run B are excerpts (README now says so) |
| Literature | arXiv API (8 queries), OpenAlex (citing works, 2 searches), zbMATH, one web search | no prior quantitative lower bound for c*(d) found |

**Corrections required by the third run**, all applied:
1. Abstract and §1.3: E V is within a polynomial factor of (E V^2)^(1/2) "for products of copies of a fixed body"
   (resp. "for such products"), not "for product bodies" (for general product bodies this is the open question of §7.2).
2. §8, What was read: the published Appendix numbers (A1), Theorem A.2, Lemma A.3, Proposition A.4 are (10),
   Theorem 18, Lemma 19, Proposition 20 in arXiv v1; only Lemma 4 and Theorems 1, 3, 6, 7 agree in both versions.
3. Reichenwallner–Reitzner is cited by section ([RR16, Section 1]) instead of the arXiv page number, and "in print"
   was dropped there (the arXiv version, which carries the journal reference, was read).
4. §7.3: "A better lower bound for γ_V, that is, a pair with a larger ratio per dimension, would improve (3)" (γ_V is
   a fixed number).
5. The package README states that two recorded outputs of run B are excerpts of the full output.
6. This run is recorded in the Verification paragraph of the note, in this report and in `independent_run_2/`.
Minor changes made at the same time: B^d is named in the abstract; Lemma 2.1 notes that Rademacher states it for
uniform points; the proof of Corollary 1.3 uses c = log 1.01392; Remark 6.5 lists the fourth Monte Carlo estimate.

## Relation to the literature, novelty and scope
- **Read for the note.** The source (publisher's PDF, pp. 518–520, and pp. 514–515 for the variance conjecture);
  Meckes's problem list; Rademacher, arXiv:1008.3944v1 completely (sha256
  `13d44d569b23d7f38ab6a6c6c411aa05e1d3a143b3a47c2c7e13321289d49b64`) and the typeset published version on the
  author's web page (math.ucdavis.edu/~lrademac/monotonicity.pdf, sha256
  `6ad62cfc13bf7b504f2ad0cb488b791ebd9cb4ec1550291df1e1801a7c019f64`; header "Mathematika 58 (2012) 77–91",
  Appendix A with Proposition A.4); Klartag–Lehec arXiv:2412.15044v1, Section 1 (Theorem 1.2, normalization (2),
  remark on p. 3); the introductions of Reichenwallner–Reitzner (arXiv:1601.07295v1: "the weak conjecture is an
  equivalent formulation, see e.g. [6]"), Kunis–Reichenwallner–Reitzner (arXiv:1612.01893v1) and Reichenwallner
  (arXiv:1702.05943v2, the published open version). Not read: Groemer 1974, Miles 1971, Schneider–Weil, Klartag
  2006 (statements taken from Rademacher's Theorems 6, 7, A.2), Kingman 1969 (abstract only, from Crossref).
- **Searches (2026-10-15).** arXiv API (five queries: random simplex / random simplices with monotonicity,
  Meckes with conjecture, covariance matrix of convex bodies under inclusion, expected volume of random
  simplex); OpenAlex works citing Rademacher 2012 (13), Reichenwallner–Reitzner 2016 (4), Kunis–Reichenwallner–
  Reitzner (1), Klartag–Lehec (19), titles screened; zbMATH (keyword search, citing documents); one web search.
  The verification runs searched further (arXiv, OpenAlex, zbMATH, Semantic Scholar, several web searches); the
  third run repeated the searches on 2026-10-15 (arXiv API, OpenAlex, zbMATH, one web search).
  No quantitative lower bound for c*(d), no statement of its exponential growth and no product amplification of
  non-monotonicity was found. A search that finds nothing is not a proof of novelty; no priority is claimed.
- **Credit.** The question is Meckes's, as recorded by Reitzner; the equivalence with slicing is Rademacher's
  (also stated by Reichenwallner–Reitzner); slicing was solved by Klartag and Lehec; Groemer's theorem, Miles's
  formula and Kingman's closed form are classical and cited.
- **Scope.** Q1: answered in print, explicit self-contained version added. Q2: the exponential order of c*(d)
  is established (1.01392 ≤ liminf c*(d)^(1/d) ≤ limsup c*(d)^(1/d) < ∞); the explicit bound is trivial for
  d ≤ 6524; the exact value of c*(d) and the limit of c*(d)^(1/d) remain open.

## 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/