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Isomorphic Busemann--Petty for arbitrary measures: the sharp order

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Pith's one-line read The optimal constant in the isomorphic Busemann–Petty problem for arbitrary even densities grows exactly like the square root of the dimension: c√n ≤ C_n ≤ √n.

desk verdict The paper proves the sharp C_n ~ sqrt(n) lower bound for the two-body Busemann–Petty problem; the new geometric lemmas are solid, and the only real soft spot is the black-box reliance on Klartag–Livshyts. read the letter →

arxiv 2608.02098 v1 pith:6BT75TS3 submitted 2026-08-03 math.FA

Alexander Koldobsky , Artem Zvavitch This is my paper · ORCID
classification math.FA MSC 52A2052A4052A2360D05
keywords Busemann-PettyproblemarbitrarymeasureshyperplanesectionsconvexbodiesisomorphicconstantrandomroundingGaussianmixturesslicinginequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the optimal constant in the isomorphic Busemann–Petty problem for arbitrary even densities grows exactly like the square root of the dimension. Earlier work had shown the constant is at most $\sqrt{n}$; this paper proves a matching lower bound $c\sqrt{n}$, so the order is sharp. The proof combines a one-scale construction that gives $\sqrt{n/\log n}$ with a sharpening using a random-rounding construction as a black box and a spherical-averaging support-separation argument. If correct, it settles the dimensional dependence of this two-body comparison problem, in contrast with the volume case where the analogous constant is dimension-free.

What carries the argument

The argument rests on three mechanisms. A support-separation lemma converts a density with uniformly small central hyperplane integrals into a compactly supported probability density whose support lies in an outer body $T$ but outside a Euclidean ball $L$, using spherical averaging of the central Radon transform. An annular comparison density $h$, supported in $L \setminus T$, has central sections bounded below by an absolute constant (or by $1/v$ in the one-scale version). A gluing principle combines these two densities with a small Gaussian perturbation to enforce strict positivity, producing a two-body comparison with section inequalities in one direction and total-mass inequality in the other. The sharp l

What would settle it

A reader could falsify the sharp lower bound by checking the constants in the quoted random-rounding construction: if the body $T$ has volume radius growing faster than a constant or the density $g$ has central sections larger than $a_0/\sqrt{n}$, then Proposition 6.1 collapses and only the one-scale $\sqrt{n/\log n}$ bound would follow. More directly, exhibiting any even continuous strictly positive density and two origin-symmetric convex bodies satisfying the hyperplane-section inequalities with total-mass ratio smaller than $c\sqrt{n}$ for every absolute $c>0$ would refute the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: there is an absolute constant $c>0$ such that for every $n\ge 2$, $c\sqrt{n} \le C_n \le \sqrt{n}$, where $C_n$ is the smallest factor such that for every even, continuous, strictly positive density $f$ and every pair of origin-symmetric convex bodies $K,L$, hyperplane-section inequalities $\int_{K\cap \xi^\perp} f \le \int_{L\cap \xi^\perp} f$ for all $\xi$ imply $\int_K f \le C_n \int_L f$. The paper proves the lower bound by constructing explicit witness bodies and densities. The sharp example uses a body $T$ with bounded volume radius and an even probability density $g$ whose central hyperplane integrals are $O(1/\sqrt{n})$; a new lemma shows that small central sections force most of the mass of such a density to lie away from any ball of ra

Load-bearing premise

The sharp lower bound depends on the existence, for every large dimension, of an origin-symmetric body with volume radius bounded by an absolute constant and of an even probability density that gives a fixed positive mass to the body while having all central hyperplane integrals at most a constant times $1/\sqrt{n}$; this is asserted from a cited random-rounding construction and not proved in this paper.

Editorial extensions

If this is right

  • The optimal constant for arbitrary even densities is dimensionally order √n; no dimension-free bound is possible in this setting.
  • The witness bodies and densities show that continuous strictly positive even densities cannot be replaced by log-concave ones in an order-√n result, as the paper notes explicitly.
  • The one-scale lower bound √(n/log n) follows from a complete, self-contained geometric construction and identifies the core mechanism; the sharp bound needs the stronger random-rounding input.
  • The spherical-averaging identity that converts small central sections into mass decay away from the origin has independent uses in comparing full-dimensional and lower-dimensional integrals of densities.
  • The sharp example produces a density that is not log-concave, delineating the boundary beyond which the isomorphic constant must grow with dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The support-separation lemma suggests a general principle: a density with all central hyperplane sections O(n^{-1/2}) must be spread out, so any convex body with bounded volume radius that carries such a density must have most of its mass at distance ≳√n from the origin—this may transfer to other geometric tomography questions.
  • The gap between the one-scale bound (√(n/log n)) and the sharp bound (√n) is exactly the volume-radius gap: replacing the one-scale body's √log n volume radius by an absolute constant yields the sharp constant, so any improvement in the volume-radius of such random constructions would directly sharpen such two-body comparisons.
  • A natural testable extension is whether the same sharp order holds for even densities that are not strictly positive or for non-symmetric densities; the gluing step uses strict positivity only via a small Gaussian term, so a limiting argument may extend the result to the closure of this class.
  • The non-log-concavity of the witness density clarifies that the log-concave Busemann–Petty constant remains dimension-free; any attempt to improve the arbitrary-measure constant must leave the log-concave class.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies the optimal constant C_n in the isomorphic Busemann–Petty problem for arbitrary even, continuous, strictly positive densities on R^n. The main result, Theorem 1.2, asserts c√n ≤ C_n ≤ √n, thereby determining the sharp order. The upper bound was previously proved by the authors. For the lower bound, the paper first gives a complete one-scale construction (Theorem 1.3) yielding C_n ≥ c√(n/log n), using Gluskin polytopes, Gaussian mixtures, and a support-separation/gluing principle. It then upgrades this to the sharp c√n lower bound by importing the Klartag–Livshyts random-rounding construction as a black box (Proposition 6.1) and combining it with a spherical-averaging support-separation lemma (Lemma 6.2, Proposition 6.3). All in-paper estimates are proved with explicit absolute constants; the sharp part is logically downstream of Proposition 6.1.

Significance. If the black-box input is valid, the paper settles the sharp order of the isomorphic Busemann–Petty constant for arbitrary measures, matching the known upper bound up to an absolute factor. The proof introduces clean, reusable support-separation and gluing tools, and the one-scale bound is fully self-contained and independent of [14]. The paper is transparent about the black-box nature of Proposition 6.1. The main caveat is that the c√n lower bound rests entirely on an unproved consequence of the proof of [14, Theorem 1.1]; if that consequence fails, only the c√(n/log n) result would survive.

major comments (2)
  1. [Section 6, Proposition 6.1] Proposition 6.1 is the sole external input for the sharp lower bound, yet its proof is only a reference to Steps 1–3 of [14, proof of Theorem 1.1]. The manuscript does not reproduce the construction or verify all the listed properties. Since Theorem 1.2 is entirely downstream of this proposition, please either include a complete proof of Proposition 6.1 in the paper, or state it as a formally quoted theorem of [14] with explicit page/equation references and demonstrate that the constants behave as asserted. In particular, check (27)–(29) and the evenness, smoothness, and probability normalization of g. If any of these properties fails or requires a different scaling, the c√n lower bound collapses, although the one-scale Theorem 1.3 would remain valid.
  2. [Section 6, Proof of Proposition 6.1] The sentence 'Their density is the convolution of the standard Gaussian density with an even discrete probability measure' is asserted without derivation. This evenness is load-bearing: Proposition 6.3 and the gluing principle require the densities p, h, and f to be even. Please either define the discrete measure explicitly or cite the precise location in [14] where its evenness is established. Similarly, the notation 'T = C6K' is ambiguous and should be written as C_6 K with the meaning of C_6 specified.
minor comments (3)
  1. [Lemma 2.4] Statement typo: the correct bounds are c/√n ≤ Eφ(n⟨Θ,ξ⟩) ≤ C/√n, as the proof establishes. The printed statement c√n ≤ Eφ(...) ≤ C√n is inconsistent with φ ≤ 1 and could mislead a reader. The later use in Proposition 2.7 correctly uses the c/√n lower bound.
  2. [Section 6, Lemma 6.2] Identity (31) is derived only in prose. A short display-level derivation, or a reference to a standard spherical-Radon-transform formula, would improve verifiability.
  3. [References] Reference [14] lists the corrected arXiv version. If the published version in the GAFA 2020 lecture notes is the primary source, cite it with the page range and any corrigendum.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sharp lower bound is built from new gluing and support-separation lemmas plus an external Klartag–Livshyts black box; self-citations are either reproved in full or are independent prior theorems.

full rationale

Walked the derivation chain. The upper bound C_n ≤ sqrt(n) in Theorem 1.2 is imported from the authors' earlier paper [21], but that is a parameter-free theorem with an independent proof and does not assume the lower bound; it is independent support, not a circular input. The one-scale lower bound (Theorem 1.3) is proved in Sections 2–5 with complete proofs of the random-direction selection (Proposition 2.7), the Gaussian marginal estimate (Lemma 2.9), the annular comparison density (Proposition 4.2), and the gluing principle (Proposition 4.3). The paper explicitly says it 'include[s] complete proofs' of the concentration and random-selection arguments, using only Gluskin's classical polytope-volume estimate as a non-elementary volumetric input. The sharp lower bound rests on Proposition 6.1, stated as a black-box consequence of Klartag and Livshyts [14], which has no author overlap with the present paper; the manuscript even stresses that Theorem 1.2 is not a formal consequence of the slicing example in [14] and adds a spherical-averaging support-separation lemma (Lemma 6.2), continuous cutoffs (Proposition 6.3), and the gluing principle. Nothing in Propositions 4.2 or 4.3 assumes the section-comparison or total-mass ordering; their hypotheses are purely geometric support and volume-radius bounds. There is a statement/proof mismatch in Lemma 2.4 (the statement displays c sqrt(n) and C sqrt(n), while the proof gives c/sqrt(n) and C/sqrt(n)), but this is a typo, not a circular definition, and Proposition 2.7 uses the proof's bound. No claimed prediction reduces by construction to its inputs, so there is no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on two imported theorems (Gluskin's polytope-volume estimate and the Klartag-Livshyts black box) plus standard facts about ball volumes, sphere nets, and Gaussian marginals that the paper proves or cites. No free parameters are fitted: the constants in the construction are absolute and existential, not tuned to data. No new entities are introduced.

assumptions (3)
  • standard math Gluskin's polytope-volume estimate (Theorem 2.1): the convex hull of M points of B_2^n has volume at most (C/n)√(log(1 + M/n)) per unit ball.
    Sole non-elementary volumetric input of the one-scale part; cited from Gluskin [11] and Pisier [24] without proof. Used in Lemma 2.8 to bound |P|^{1/n} ≤ C√(log n)/n.
  • domain assumption Klartag-Livshyts random-rounding black box (Prop 6.1): for every large n there exist T and g with r0√n B_2^n ⊂ T, |T|^{1/n} ≤ C0, ∫_T g ≥ c0, and ∫_{ξ⊥} g ≤ a0/√n for every ξ.
    The sharp lower bound stands or falls on this external theorem. The paper asserts (27)-(29) follow from Steps 1-3 of [14, Theorem 1.1] and does not reproduce them; this is a disclosed reliance on prior literature.
  • standard math Euclidean nets of the sphere of size (3/δ)^n (Lemma 2.6) and standard ball volume and surface-area estimates (Lemma 2.2).
    Background volumetric facts used in the net argument of Prop 2.7 and throughout; Lemma 2.6 is cited to Pisier, Lemma 2.2 is proved via Stirling-type gamma bounds.

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Pith. "Pith review of Isomorphic Busemann--Petty for arbitrary measures: the sharp order." pith.science (2026). https://pith.science/paper/6BT75TS3

@misc{pith2026260802098,
  author       = {Pith},
  title        = {Pith review of: Isomorphic Busemann--Petty for arbitrary measures: the sharp order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BT75TS3}},
  note         = {Machine review of arXiv:2608.02098}
}
abstract

Let $C_n$ be the optimal constant with the following property. For every even, continuous, strictly positive density $f$ on $R^n$ and all origin-symmetric convex bodies $K,L\subset R^n$, the inequalities $$ \int_{K\cap\xi^\perp}f \leq \int_{L\cap\xi^\perp}f \qquad\text{for all }\xi\in S^{n-1} $$ imply $\int_Kf\leq C_n\int_Lf$. In an earlier paper the authors proved that $C_n\leq\sqrt n$. In this paper, we prove the matching lower bound $C_n\geq c\sqrt n$. To simplify the exposition, we first give a complete one-scale construction, based on earlier work of Klartag and Koldobsky, which yields $C_n\geq c\sqrt{n/\log n}$. For the sharp result, we use the random-rounding construction of Klartag and Livshyts as a black box and combine it with a spherical-averaging support-separation argument.

Figures

Figures reproduced from arXiv: 2608.02098 by the authors.

Figure 1
Figure 1. Schematic representation of the support-separation and gluing procedure inside a central hyperplane ξ ⊥. The density p is supported in T \ L and has small central hyperplane integrals, whereas h is supported in L \ T and has uniformly large central hyperplane integrals [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A schematic two-dimensional section of the annular construction. The light-gray region is U \ int((1 + 1/n)Q), which contains the support of h0. The dark-gray region is (1 − 1/n)U \ (1 + 2/n)Q, where h0 = 1. We note that the body Q and the Euclidean ball U need not be nested. Next, h0 = 1 on  1 − 1 n  RBn 2 \  1 + 2 n  Q. Consequently, (21) Z ξ⊥ h0 ≥ |B n−1 2 | 1 − 1 n  R n−1 −  1 + 2 n n−1 |Q ∩ ξ ⊥|. We c… view at source ↗

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