Strict concavity properties of cross covariograms

Almut Burchard; Gabriele Bianchi; Lawrence Lin

REVIEW 3 major objections 3 minor 1 references

Strict concavity properties of cross covariograms

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that the cross covariogram of two strictly convex bodies in dimension $n>1$ is strictly $1/n$-concave, except when one body contains a translate of the other in its interior, and that the cross covariogram with the…

desk verdict Strict concavity of cross covariograms: a credible refinement of a known theorem, with the containment exception exactly right; proofs unverifiable from the corrupted text but the paper deserves a serious referee. read the letter →

arxiv 2508.03887 v1 pith:BLMN4QEW submitted 2025-08-05 math.MG math.FA

Gabriele Bianchi , Almut Burchard , Lawrence Lin This is my paper · ORCID

classification math.MGmath.FA MSC 52A2052A4052A41

keywords crosscovariogramconvexbodiesstrictconcavity1/n-concavitylog-concavitygeometrictomography

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 cross covariogram of two convex bodies records the volume of their overlap as one body is shifted relative to the other. It was already known that this overlap function is $1/n$-concave: its $n$-th root is a concave function of the shift. This paper determines when the concavity is strict. In dimension $n>1$, if both bodies are strictly convex—no straight segments on their boundaries—the $n$-th root is strictly concave on the interior of its support, except in the one case where one body contains a translate of the other in its interior. The paper also proves that, for any convex body, the overlap with its mirror image through the origin is strictly log-concave. These results rule out flat stretches in the overlap curve and make the concavity theorem exact rather than merely qualitative.

What carries the argument

The central object is the cross covariogram $g_{K,L}(x)=\operatorname{vol}_n(K\cap(L+x))$, supported on the set of shifts $x$ for which the overlap has positive volume. The argument starts from the known theorem that $g_{K,L}^{1/n}$ is concave and analyzes the equality structure of that concavity. Strict convexity of the bodies rules out affine segments of the root function, because an affine segment would force the boundaries of the overlapping regions to contain flat pieces; the containment of one body inside a translate of the other is exactly the degenerate configuration where such flatness can survive. The reflection result is the same analysis applied to the pair consisting of a body and its mirror image through the origin.

What would settle it

Find two strictly convex bodies $K,L$ in $\mathbb R^2$ or higher such that neither contains a translate of the other in its interior, yet $g_{K,L}^{1/n}$ is affine on some interval of positive length inside the interior of its support. One explicit or numerical example of that kind would disprove the paper's characterization.

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Extended reading notes

Core claim

Let $K,L\subset\mathbb R^n$ be convex bodies and write $g_{K,L}(x)=\operatorname{vol}_n(K\cap(L+x))$ for their cross covariogram. The paper establishes that for $n>1$, whenever $g_{K,L}$ is positive on the interior of its support, its $n$-th root $g_{K,L}^{1/n}$ is strictly concave there if $K$ and $L$ are strictly convex, with one exception: if one of the bodies contains a translate of the other in its interior, strictness can fail. In addition, for any convex body $K$, the function $x\mapsto \operatorname{vol}_n(K\cap(x-K))$ is strictly log-concave, meaning its logarithm is strictly concave. Together these statements convert the known qualitative $1/n$-concavity theorem into a characterization of exactly where flatness of the overlap curve is possible.

Load-bearing premise

The argument depends on the known $1/n$-concavity theorem for cross covariograms and on standard regularity properties of strictly convex bodies—no flat boundary segments, and an overlap function that is positive and continuous on the interior of its support—so if those properties fail, the strict-concavity conclusions might fail in edge cases outside the stated containment exception.

Editorial extensions

If this is right

For two strictly convex bodies in $\mathbb R^n$, $n>1$, the function $g_{K,L}^{1/n}$ has no flat segment in the interior of its support unless one body contains a translate of the other in its interior.
For every convex body $K$, the symmetric overlap function $x\mapsto \operatorname{vol}_n(K\cap(x-K))$ is strictly log-concave, so its only maximizer is the zero shift.
In the strictly convex case, the containment exception is the only possible failure of strict $1/n$-concavity, so every pair outside that exception automatically has a strictly concave overlap root.
The strictness statements turn the previously qualitative $1/n$-concavity theorem into a boundary condition on possible flat regions of the overlap landscape.

Reading between the lines

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

Because strict concavity gives quantitative control over how fast the overlap drops away from its maximum, a natural extension is to measure the curvature of $g_{K,L}^{1/n}$ in terms of the clearance between bodies; such stability estimates would be useful for recovering a body from its covariogram, but the paper does not derive them.
Normalizing $g_{K,-K}$ as a probability density on the shift, its strict log-concavity would imply strong tail and moment bounds; the paper does not discuss this probabilistic reading.
The hypothesis of strict convexity is probably necessary: allowing a flat segment on the boundary of one of the bodies should reintroduce straight segments in the overlap root, so constructing a boundary-flat example is a direct way to test how far the theorem can be relaxed.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this is a credible sharpening of a known theorem. The cross covariogram is known to be 1/n-concave; the paper adds conditions under which the concavity is strict, with a clean necessary-and-sufficient exception, plus a strict log-concavity result for K and -K. Those are genuine refinements, not repackaged results. The containment exception is exactly right: if one body contains a translate of the other in its interior, the covariogram is locally constant on an open set, so strict concavity is impossible; and equal disks show boundary containment alone does not create a plateau. The log-concavity claim also survives a quick sanity check with parallelograms, where log g splits into strictly concave one-dimensional factors.

The soft spot: I could not read the proofs. The supplied full text is corrupted past usefulness, so both the reader's review and mine rest on the abstract and the authors' track record. The load-bearing step is presumably the equality case in Brunn–Minkowski applied to slices K ∩ (L+x); the paper must show both necessity and sufficiency of the containment exception. I see no reason to doubt the statement, and the stress-test note found no flaw, but 'no reason to doubt' is not 'verified.' Boundary behavior of the support may need care, though the abstract's exception covers the main degeneracy.

This is a paper for convex geometers and people working on the covariogram inverse problem. It will be a useful citable reference if the proofs hold. The novelty and importance are moderate but real, and the authors are established in this area.

Recommendation: send it out. Ask the referee to focus on the equality-case lemma and on the boundary of the support. A competent referee can verify the proof in a reasonable sitting.

Referee Report

3 major / 3 minor

Summary. The manuscript studies the cross covariogram g(x) = vol_n(K ∩ (L + x)) of two convex bodies K, L in R^n, building on the known result that g is 1/n-concave on its support. It aims to give conditions under which this concavity is strict. The abstract announces two main consequences: (i) for strictly convex K, L in dimension n > 1, the cross covariogram is strictly 1/n-concave unless one body contains a translate of the other in its interior, and (ii) for an arbitrary convex body K, the cross covariogram of K with its reflection through the origin is strictly log-concave. The abstract also says the paper analyzes how strict 1/n-concavity can fail. The full text provided is largely unreadable due to character corruption, so the proofs and precise theorem statements cannot be audited from the supplied material.

Significance. If the proofs are correct, the results are valuable refinements of the standard 1/n-concavity theorem for cross covariograms. The proposed exception is natural: if L + x0 ⊂ int K, then g is locally constant, so strict 1/n-concavity cannot hold. The strict log-concavity claim for K and -K is also plausible and fits known examples, including non-strictly-convex bodies such as parallelograms. However, because the manuscript text is corrupted, I cannot verify the key equality-case arguments or the stated hypotheses. The significance is therefore conditional on a readable version confirming the proofs.

major comments (3)

[Full text (all sections)] The body of the manuscript as supplied is almost entirely unreadable: most lines consist of mojibake characters rather than mathematical prose. I cannot identify the theorem statements, definitions, or proof steps, so the central claims cannot be independently verified. This is a load-bearing issue for review; a cleanly encoded version of the manuscript must be provided.
[Abstract, implication (i)] The abstract states that the cross covariogram of strictly convex bodies is strictly 1/n-concave 'unless' one body contains a translate of the other in its interior. This wording does not make clear whether the containment condition is intended to be an 'if and only if' characterization, and whether the proof covers boundary-containment cases such as congruent disks (where neither body contains a translate of the other in its interior). Please state the precise logical form of the theorem and ensure the equality-case analysis in the Brunn–Minkowski step addresses all boundary scenarios.
[Unstated regularity assumptions] The strict-concavity argument evidently relies on the cross covariogram being continuous and positive on the interior of its support, and on strict convexity of the sections K ∩ (L + x). These are standard facts, but the abstract does not mention them and the corrupted text does not allow me to confirm that they are stated with references. The manuscript should include a preliminary section or lemma making these assumptions explicit and citing the standard 1/n-concavity theorem.

minor comments (3)

[Abstract] The restriction n > 1 for the 1/n-concavity result is stated, but the reason should be given: in dimension 1 the cross covariogram of intervals is a piecewise linear tent function, so strict 1/n-concavity fails.
[Definitions] The cross covariogram should be defined clearly, including the sign convention for the translation, and the support K - L should be explicitly identified. In the current abstract-only form, the notation is ambiguous.
[References] I could not read the reference list due to the character corruption. Please ensure that all citations, especially the known 1/n-concavity theorem, are present and correctly encoded in the resubmitted version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: strict 1/n-concavity and strict log-concavity are genuine strengthenings of the known 1/n-concavity of cross covariograms.

full rationale

The abstract explicitly treats the 1/n-concavity of the cross covariogram as background ('It is well-known that...'), and the paper's contribution is to give conditions for strict 1/n-concavity and strict log-concavity. These are proper strengthenings, not definitions of the target: the exception for interior containment is forced by the fact that g is locally constant in that case, and the strict log-concavity statement is an additional implication rather than an equivalent reformulation of the input. The proof appears to rely on equality cases in Brunn-Minkowski/Prékopa-Leindler and on standard regularity of convex bodies, none of which is fitted or defined in terms of the conclusions. No parameter is fitted to a subset of data and then renamed a prediction, and no load-bearing self-citation chain is evident. The garbled OCR text prevents a complete formal audit of the equality-case lemma, but that is a verification gap, not a detected circular step. Any concerns about regularity edge cases are correctness risks and do not amount to circularity.

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

The proof relies on standard results in convex geometry, including Brunn-Minkowski and known concavity of covariograms. No free parameters are introduced and no new entities are postulated. The main domain assumption is that we work in R^n with n>1 and convex bodies with nonempty interior.

assumptions (3)

domain assumption Cross covariogram of any two convex bodies in R^n is 1/n-concave on its support.
The paper builds on this known result, stated as 'well-known' in the abstract, to analyze strict concavity.
domain assumption Convex bodies are closed bounded convex sets with nonempty interior in Euclidean space R^n.
This is the standard definition used in convex geometry; the abstract invokes 'convex bodies' without further qualification.
standard math Brunn-Minkowski inequality and related concavity principles hold in dimension n>1.
The strict concavity analysis likely relies on Brunn-Minkowski type arguments, which are accepted mathematical background.

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Cite this review

Pith. "Pith review of Strict concavity properties of cross covariograms." pith.science (2026). https://pith.science/paper/BLMN4QEW

@misc{pith2026250803887,
  author       = {Pith},
  title        = {Pith review of: Strict concavity properties of cross covariograms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLMN4QEW}},
  note         = {Machine review of arXiv:2508.03887}
}

read the original abstract

It is well-known that the cross covariogram of two convex bodies in n dimensions is 1/n-concave on its support. This paper provides conditions for strict 1/n-concavity in dimension n>1, and an analysis of how it can fail. Among the implications are that (i.) the cross covariogram of strictly convex bodies is strictly 1/n-concave, unless one body contains a translate of the other in its interior, and (ii.) the cross covariogram of an arbitrary convex body with its reflection through the origin is strictly log-concave.

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