# Reading roadmap: A product counterexample to the simplex maximum for projection-body volume The argument computes projection-body volumes from facet vectors. An orthogonal Cartesian product makes the normalized volume multiplicative, while a simplex projection body is a scaled cube plus one diagonal segment. These identities reduce a product of two ten-dimensional simplices to an integer comparison with the twenty-dimensional simplex benchmark. Repeating ten-dimensional factors yields an exponential excess in all sufficiently large dimensions. Source SHA-256: `628bd94e51987a4ab5879c572a01837154b2ace52c1be9fe9d672178951541df`. Indexed 20,710 characters. Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates. ## Set the benchmark and scope Define projection-body volume normalized by body volume, state the simplex benchmark and earlier results, and announce an explicit product witness and a consequence for large dimensions. Claims: thm:counterexample ## Obtain the geometric identities Represent polytopal projection bodies by facet area-normal segments, separate the facets of an orthogonal Cartesian product, and track area-normal vectors under affine maps. Claims: lem:facets, prop:product, lem:affine ## Compute the simplex benchmark Write the standard simplex projection body as a scaled cube plus one diagonal segment; compute its volume by extending parallel fibers, then apply affine covariance to other simplices. Claims: prop:simplex ## Compare products with simplices Combine product multiplicativity and the simplex value for two ten-dimensional factors. Repeat such factors and add one simplex to reach every residue class, comparing their values to a fixed exponential upper estimate for the simplex benchmark. Claims: thm:counterexample, cor:exponential-excess ## Internal claim index ### Theorem 1 — `thm:counterexample` Source lines 90–101. Product multiplicativity gives the square of the ten-dimensional simplex value. Dividing by the twenty-dimensional simplex benchmark reduces the comparison to a binomial coefficient and an explicit strict integer inequality. - `prop:product` (proof_dependency; source_audit_annotation): The named theorem proof invokes both propositions to express the product value as c_10 squared. - `prop:simplex` (proof_dependency; source_audit_annotation): The named theorem proof invokes both propositions to express the product value as c_10 squared. Candidate references: `eq:proposed-bound` (internal_reference); `prop:product` (candidate_proof_dependency); `prop:simplex` (candidate_proof_dependency); `eq:proposed-bound` (candidate_proof_dependency). ### Lemma 2 — `lem:facets` Source lines 153–167. Project each facet with Jacobian equal to the absolute normal component. Almost every projected point is covered by an entry facet and an exit facet, producing the factor one half; segment support functions give the zonotope representation. Candidate references: `eq:facet` (candidate_proof_dependency); `eq:zonotope` (candidate_proof_dependency). ### Proposition 3 — `prop:product` Source lines 193–205. The facets of an orthogonal product have one facet factor and one whole factor. Their area-normal vectors give two separate support-function sums, hence a product projection body; volume scaling then yields multiplicativity of the normalized functional. - `lem:facets` (proof_dependency; source_audit_annotation): The product proof applies the facet support-function formula to its two facet families. Candidate references: `lem:facets` (candidate_proof_dependency); `eq:product-body` (candidate_proof_dependency). ### Lemma 4 — `lem:affine` Source lines 245–254. For polytopes, the inverse transpose transforms facet normals and the determinant scales facet areas. Apply the segment representation and cancel volume factors; approximate arbitrary convex bodies by inscribed polytopes to pass the identities to the limit. - `lem:facets` (proof_dependency; source_audit_annotation): The affine proof applies the zonotope representation to the transformed area-normal vectors. Candidate references: `eq:zonotope` (candidate_proof_dependency). ### Proposition 5 — `prop:simplex` Source lines 311–321. Compute the simplex facet area-normal vectors to obtain a scaled translate of a cube plus its diagonal segment. Each parallel fiber length increases by the segment length; the cube facet formula gives volume d+1. Normalize by simplex volume and use affine covariance for all simplices. - `lem:facets` (proof_dependency; source_audit_annotation): The simplex proof inserts its facet generators into the centered-segment representation. - `lem:affine` (proof_dependency; source_audit_annotation): Affine invariance transfers the standard simplex value to every full-dimensional simplex. Candidate references: `lem:facets` (candidate_proof_dependency); `eq:facet` (candidate_proof_dependency); `fig:extrusion` (candidate_proof_dependency); `lem:affine` (candidate_proof_dependency). ### Corollary 6 — `cor:exponential-excess` Source lines 423–432. For each dimension, use repeated ten-dimensional simplices and one simplex of dimension ten through nineteen. Bound the simplex benchmark by (n+1) times (11/4)^n, compare this with the block value, and choose a common threshold across the ten residue classes. - `prop:product` (proof_dependency; source_audit_annotation): The corollary proof iterates the product identity and substitutes the simplex values for all factors. - `prop:simplex` (proof_dependency; source_audit_annotation): The corollary proof iterates the product identity and substitutes the simplex values for all factors. Candidate references: `prop:product` (candidate_proof_dependency); `prop:simplex` (candidate_proof_dependency). ## External inputs and references ### [1] Brannen — `external:Brannen` Source lines 488–494. N.~S.~Brannen, Volumes of projection bodies, Mathematika 43 (1996), no.~2, 255--264. \href{https://doi.org/10.1112/S002557930001175X}{doi:10.1112/S002557930001175X}. The simplex maximum conjecture motivating the comparison. Role: **background**. Expanded native manuscript lines 63–68; all six numbered claims use locally supplied arguments. - `section:1` (background; source_audit_annotation): The simplex maximum conjecture motivating the comparison. Candidate references: `section:1` (external_reference). ### [2] ChenEtAl — `external:ChenEtAl` Source lines 494–500. S.~Chen, Y.~Feng, Y.~Li, D.~Xi, and L.~Xu, Petty's conjectured projection inequality in dimension three, preprint, 2026, MathSciDoc:2608.23001. \href{https://archive.ymsc.tsinghua.edu.cn/pacm_download/741/12780-pettyinequality.pdf}{Full text}. Earlier dimension-three minimum and simplex upper-bound results, cited as context. Role: **background**. Expanded native manuscript lines 76–80; all six numbered claims use locally supplied arguments. - `section:1` (background; source_audit_annotation): Earlier dimension-three minimum and simplex upper-bound results, cited as context. Candidate references: `section:1` (external_reference). ### [3] FengEtAl — `external:FengEtAl` Source lines 500–506. Y.~Feng, S.~Hu, W.~Liu, and L.~Xu, On the reverse projection inequality, preprint, 2026, MathSciDoc:2608.23002. \href{https://archive.ymsc.tsinghua.edu.cn/pacm_download/743/12781--2026.8.26.pdf}{Full text}. Earlier counterexamples in every dimension at least nine, distinguished from the present explicit product construction. Role: **background**. Expanded native manuscript lines 80–88; all six numbered claims use locally supplied arguments. - `section:1` (background; source_audit_annotation): Earlier counterexamples in every dimension at least nine, distinguished from the present explicit product construction. Candidate references: `section:1` (external_reference). ### [4] Henk — `external:Henk` Source lines 506–512. M.~Henk, Note on projection bodies of zonotopes with \(n+1\) generators, Acta Mathematica Scientia 45 (2025), 96--103. \href{https://doi.org/10.1007/s10473-025-0107-9}{doi:10.1007/s10473-025-0107-9}. A lower bound for the optimal upper constant among centrally symmetric bodies, cited as dimensional-growth context. Role: **background**. Expanded native manuscript lines 118–124; all six numbered claims use locally supplied arguments. - `section:1` (background; source_audit_annotation): A lower bound for the optimal upper constant among centrally symmetric bodies, cited as dimensional-growth context. Candidate references: `section:1` (external_reference). ### [5] Ludwig — `external:Ludwig` Source lines 512–518. M.~Ludwig, Projection bodies and valuations, Advances in Mathematics 172 (2002), no.~2, 158--168. \href{https://doi.org/10.1016/S0001-8708(02)00021-X}{doi:10.1016/S0001-8708(02)00021-X}. Classical affine covariance context; the paper supplies its own proof in Lemma 4. Role: **background**. Expanded native manuscript lines 60–63; all six numbered claims use locally supplied arguments. - `section:1` (background; source_audit_annotation): Classical affine covariance context; the paper supplies its own proof in Lemma 4. Candidate references: `section:1` (external_reference). ### [6] Petty — `external:Petty` Source lines 518–525. C.~M.~Petty, Projection bodies, in Proceedings of the Colloquium on Convexity (Copenhagen, 1965), K\o benhavns Universitet Matematiske Institut, Copenhagen, 1967, pp.~234--241. Classical affine covariance context; the paper supplies its own proof in Lemma 4. Role: **background**. Expanded native manuscript lines 60–63; all six numbered claims use locally supplied arguments. - `section:1` (background; source_audit_annotation): Classical affine covariance context; the paper supplies its own proof in Lemma 4. Candidate references: `section:1` (external_reference). ### [7] Saroglou — `external:Saroglou` Source lines 525–533. C.~Saroglou, On the shape of a convex body with respect to its second projection body, Advances in Applied Mathematics 67 (2015), 55--74. \href{https://doi.org/10.1016/j.aam.2015.03.004}{doi:10.1016/j.aam.2015.03.004}. Preprint: \href{https://arxiv.org/abs/1409.4347v2}{arXiv:1409.4347v2}. Context for the distinction between projection-body and polar functionals and for Cauchy’s facet formula, which is proved locally. Role: **background**. Expanded native manuscript lines 70–74; all six numbered claims use locally supplied arguments. - `section:1` (background; source_audit_annotation): Context for the distinction between projection-body and polar functionals and for Cauchy’s facet formula, which is proved locally. - `sec:facets` (background; source_audit_annotation): The source identifies its locally proved facet formula with the classical polytopal Cauchy formula. Candidate references: `section:1` (external_reference); `sec:facets` (external_reference). ## Coverage and limits {"mode": "native_latex_index", "claims": 6, "sections": 4, "external_references": 7, "source_bibliography_entries": 7, "unresolved_bibliography": 0, "unresolved": 0} - Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer. - Numbers reflect simple source counters; labels and line anchors are authoritative. - Extractive statements and proof openings are marked; they are not reconstructed proof gists. - Macros, included files and custom counters require an expanded source or annotated fallback.