The maximal volume of projections of the cross-polytope

Grigory Ivanov

REVIEW 3 major objections 2 minor 1 cited by

The maximal volume of projections of the cross-polytope

T0 review · 3 major / 2 minor · reviewed 2026-07-15 · grok-4.5

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Pith's one-line read Any k-dimensional orthogonal projection of the regular cross-polytope has volume at most 2^k/k!, with equality only for coordinate subspaces.

desk verdict Claims a clean proof of the sharp projection-volume bound for the cross-polytope (and a general abs-convex-hull form), via a coherent triangulation-plus-solid-angle strategy; only the abstract is available, so the key comparisons remain unchecked. read the letter →

arxiv 2607.12072 v1 pith:ZARMOWN3 submitted 2026-07-13 math.FA

Grigory Ivanov This is my paper · ORCID

classification math.FA

keywords cross-polytopeorthogonalprojectionsvolumeboundsabsoluteconvexhullGaussiansolidanglecompletefanradialtriangulationgeometry

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 proves a sharp upper bound on the volume of the absolute convex hull of any spanning family of vectors in k-dimensional space, controlled by the square root of the determinant of the sum of their outer products. As a direct consequence, every k-dimensional orthogonal projection of the regular n-dimensional cross-polytope has volume at most 2^k/k!, and the bound is attained only when the projection is onto a coordinate subspace. The argument works by triangulating the boundary of the absolute convex hull, comparing the volume contribution of each radial simplex against the Gaussian solid angle of its positive cone, and using that those cones form a complete fan covering the whole space. If the bound holds, it settles a conjectured extremal property of cross-polytope projections and supplies a clean comparison between Euclidean volume and a quadratic form built from the generating vectors. A sympathetic reader cares because the result gives an exact, equality-characterized maximum for a classical family of convex bodies under orthogonal projection.

What carries the argument

A triangulation of the boundary of the absolute convex hull into radial simplices, each compared with the Gaussian solid angle of its positive cone, together with the fact that the radial cones form a complete fan; this comparison converts local determinant estimates into a global volume bound.

What would settle it

Exhibit a spanning family of vectors in some dimension k whose absolute convex hull has k-volume strictly larger than (2^k/k!) times the square root of the determinant of the sum of outer products, or a non-coordinate k-plane onto which the regular cross-polytope projects with volume larger than 2^k/k!.

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Extended reading notes

Core claim

For every spanning family of vectors v1 through vn in R^k the k-volume of the absolute convex hull of the signed vectors is at most (2^k/k!) times the square root of the determinant of the sum of the outer products vi⊗vi; after natural normalization, equality holds precisely when the nonzero vectors form an orthonormal basis. Consequently every orthogonal projection of the regular cross-polytope onto a k-dimensional subspace has volume at most 2^k/k!, with equality only for coordinate subspaces.

Load-bearing premise

The comparison of each radial-simplex determinant with the Gaussian solid angle of its positive cone, together with the claim that the radial cones form a complete fan without residual solid angle or overcounting, remains valid for an arbitrary spanning family.

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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

The punchline is simple: the paper asserts a complete proof of the conjectured sharp upper bound vol_k(P_E crosp^n) ≤ 2^k/k! for any k-dimensional orthogonal projection, with equality only on coordinate subspaces, and packages it inside a more general inequality for the absolute convex hull of any spanning family v_1,…,v_n in R^k:

vol_k(conv{±v_i}) ≤ (2^k/k!) √ det(∑ v_i ⊗ v_i),

equality (after normalization) precisely when the nonzero vectors form an orthonormal basis.

That general form and the equality characterization are the actual new content. The outline is clean and standard for the area: triangulate the boundary of the abs-convex hull, compare the determinant of each radial simplex against the Gaussian solid angle of its positive cone, then sum over the claim that those cones form a complete fan. If those local comparisons and the fan covering hold without residual solid angle or overcounting, the global bound drops out immediately. No free parameters, no circular fitting, and the equality cases line up with the geometry one expects.

The only soft spot is the one the reader already flagged: we have only the abstract. The det-versus-angle comparison and the completeness of the radial fan for arbitrary spanning families cannot be inspected. The stress-test note is right that this is not an internal contradiction or a hidden assumption visible in the outline; it is simply the absence of the derivation. Nothing in the stated strategy looks load-bearing-wrong.

This is for people who work on volume estimates, projections of polytopes, and the local theory of Banach spaces. A specialist who needs the sharp constant will get value from it once the proof is checked. It is important enough and formally grounded enough that a serious editor should send it to referees rather than desk-reject; the conjecture is classical and the claimed method is the right kind of method. I would not put it on the next reading-group list until the full text is out, and I would not cite the bound yet, but if the details hold it becomes a standard reference.

Referee Report

3 major / 2 minor

Summary. The manuscript claims a sharp upper bound on the k-volume of the absolute convex hull of any spanning family v_1,...,v_n in R^k: vol_k(conv{±v_1,...,±v_n}) ≤ (2^k/k!) √ det(∑_i v_i ⊗ v_i), with equality (after natural normalization) precisely when the nonzero vectors form an orthonormal basis. As a corollary, every k-dimensional orthogonal projection of the regular cross-polytope has volume at most 2^k/k!, with equality only for coordinate subspaces. The abstract outlines a proof by boundary triangulation of the absolute convex hull, a local comparison of radial-simplex determinants against Gaussian solid angles of positive cones, and a global summation that relies on the radial cones forming a complete fan.

Significance. If the argument is complete and correct, the paper settles a conjectured sharp geometric inequality for projections of the cross-polytope and supplies a clean equality characterization in terms of orthonormal bases and coordinate subspaces. The bound is parameter-free and sharp. The strategy (triangulation + det-versus-solid-angle comparison + complete-fan summation) is standard in spirit and, if executed carefully, would be of independent interest for volume estimates of absolute convex hulls. The abstract alone, however, does not allow verification of the load-bearing steps, so the significance remains conditional on the full derivation.

major comments (3)

Abstract proof outline: the global bound rests on the claim that the radial cones of the triangulated absolute convex hull form a complete fan (no residual solid angle, no overcounting) for an arbitrary spanning family. Without the full construction this covering property cannot be checked; any gap or hidden restriction on the family would leave the inequality unproved.
Abstract proof outline: the local comparison between the determinant of every radial simplex and the Gaussian solid angle of its positive cone is asserted to be valid and to produce the factor 2^k/k! after summation. The abstract supplies neither the precise inequality nor the equality cases of this comparison; both are load-bearing for the sharp constant and the equality characterization.
Abstract equality claim: equality is said to hold precisely when the nonzero vectors form an orthonormal basis (and, for projections, only for coordinate subspaces). Verification requires the full analysis of when equality propagates through the local comparison and the fan summation; that analysis is not available from the abstract alone.

minor comments (2)

Abstract only: notation for the absolute convex hull and for the Gaussian solid angle is introduced only by name; a full manuscript should fix conventions (e.g., normalization of solid angle, orientation of simplices) at first use.
Abstract: the phrase “after the natural normalization” is left undefined; the full text should state the normalization explicitly when the equality cases are formulated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; abstract-only geometric inequality with independent external notions.

full rationale

The abstract presents a sharp geometric inequality for volumes of absolute convex hulls of spanning families, with equality characterization in terms of orthonormal bases, and a consequence for projections of the regular cross-polytope. The claimed derivation outline (boundary triangulation of the absolute convex hull, comparison of radial-simplex determinants to Gaussian solid angles of positive cones, and summation over a complete fan of radial cones) invokes standard external notions from convex geometry and does not reduce the bound to a fitted parameter, a self-definition, or a load-bearing self-citation. No equations, prior-work citations, or uniqueness theorems appear in the available text that would allow a reduction of the form Eq. X = Eq. Y by construction. The result is presented as a first-principles comparison against Euclidean volume, determinants, and Gaussian solid angles. With only the abstract available, no circular step can be exhibited by quotation and reduction; the honest finding is therefore score 0 with empty steps. (Abstract-only status limits verification of correctness of the fan/completeness argument, but that is outside the circularity criterion.)

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

Pure geometric inequality in Euclidean space. No numerical free parameters are fitted. The argument relies on standard notions of volume, determinant and solid angle, plus the structural claim that the radial cones of the absolute convex hull form a complete fan. No new physical or geometric entities are introduced.

assumptions (3)

standard math Standard Euclidean k-volume, Gram determinant, and orthogonal projection in R^k
Background notions used to state both sides of the inequality.
domain assumption Gaussian solid angle of a positive cone is well-defined and can be compared with the determinant of a radial simplex
Core comparison step of the outlined proof; treated as available geometric machinery.
domain assumption The radial cones arising from a triangulation of the boundary of the absolute convex hull form a complete fan of R^k
Explicitly invoked in the abstract as the step that turns local comparisons into a global volume bound.

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Cite this review

Pith. "Pith review of The maximal volume of projections of the cross-polytope." pith.science (2026). https://pith.science/paper/ZARMOWN3

@misc{pith2026260712072,
  author       = {Pith},
  title        = {Pith review of: The maximal volume of projections of the cross-polytope},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZARMOWN3}},
  note         = {Machine review of arXiv:2607.12072}
}

abstract

We prove the conjectured sharp upper bound for the volume of an arbitrary lower-dimensional orthogonal projection of the regular cross-polytope. More generally, for every spanning family $v_1,\dots,v_n \in \mathbb{R}^k, $ we prove \[ \operatorname{vol}\nolimits_{k} \operatorname{conv} \{\pm v_1, \dots, \pm v_n\} \le \frac{2^k}{k!} \sqrt{\det\!\left(\sum_{i=1}^n v_i\otimes v_i\right)}. \] After the natural normalization, equality holds precisely when the non-zero vectors form an orthonormal basis. We triangulate the boundary of the absolute convex hull, compare the determinant of every radial simplex with the Gaussian solid angle of its positive cone, and then use that the radial cones form a complete fan. As a consequence, the volume of the projection of $\crosp^n$ onto any $k$-dimensional subspace is at most $2^k/k!$, with equality only for coordinate subspaces.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Geometry of the subgaussian body of an isotropic convex body
math.MG 2026-08 accept novelty 7.0 of 10

For any isotropic convex body in R^n, the subgaussian body has bounded volume ratio against the centroid body, sharp mean width O(√log n), and an orthonormal basis with subgaussian constants O(√log n).

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Reviewed July 15, 2026 · model on record in the stance chip above.