{"id":"03fbf2a7-5827-4730-92cc-fb63c60fb5ca","arxiv_id":"1908.03196","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"This paper reviews which volume functions determine a convex body, lists open minimum-volume problems in three dimensions, and announces without proof an equivalence between two conjectures for a future paper.","lead":"This survey gathers classical and recent results about which volume measurements of a convex shape reveal the whole shape. It maps open problems in three-dimensional geometry that a computer search might help settle.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved announced equivalence between the translative constant-volume property and Petty's polar projection problem is the main gap; the derivation is short and should be included.","rationale":"I read the paper as a survey whose central technical anchor is Lemma 1 and whose novel claim is the equivalence between Conjecture 5 and Petty's polar projection problem. Lemma 1 is correct: conv(K, K+t) = K + [0,t], and slicing perpendicular to t gives volume vol(K) + |t| times vol_{n-1}(K|t^perp). The subsequent reduction of the convex-hull function to volume plus brightness is sound. The main soft spot is exactly the announced equivalence, which the manuscript itself flags as deferred to a forthcoming paper. This is not an internal inconsistency; it is an unsupported load-bearing assertion. The reader's CONDITIONAL verdict is appropriate: the paper should either insert the short proof or soften the parenthetical. I do not see a separate mathematical falsehood that would warrant rejection.","tokens_in":671,"tokens_out":933,"duration_ms":161930,"concrete_test":"Derive the two reductions independently: first show that the translative constant-volume property is equivalent to b_K(u) times rho_K(u) being constant for all u; then show that Petty's condition PI_K^circ = lambda K is equivalent to the same relation. If both derivations go through, replace the 'forthcoming paper' parenthetical with a short lemma in the text. If either direction fails, the claimed equivalence is false and the survey's framing should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised unification appears in Section 3 immediately after Conjecture 6: the polar projection problem 'is equivalent to the conjecture on translative constant volume property', with only the parenthetical 'We will prove it in a forthcoming paper.' No proof, sketch, or reference is given. This is load-bearing because this sentence is what makes the translative constant-volume conjecture a modern form of Petty's old problem; without it, the survey's novel angle rests on an unsupported assertion. The good news is that the equivalence is short: Eq. (12) together with the touching-pair condition reduces the translative constant-volume property to b_K(u) times rho_K(u) being constant for all unit u, while h_PI_K = b_K and rho_(PI_K)^circ(u) = 1/h_PI_K(u) reduce the condition PI_K^circ = lambda K to the same relation. Thus the claim is likely true but should be proved in the manuscript or explicitly labelled as a conjecture. The same section's planar example is also under-specified, but the existence of non-congruent constant-width bodies with equal area makes the underlying claim credible.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This survey collects and connects results about volume functions attached to a convex body: the covariogram, the width function, the brightness function, the projection body, and the more recently studied convex-hull function. The main technical content is Lemma 1, which states that the convex-hull function along a line through the origin is determined by volume and brightness, and Theorem 5, which constructs non-congruent prisms with equal brightness and equal volume from planar bodies with equal width and area. The paper also surveys open problems, including the Bonnesen-Fenchel conjecture on Meissner tetrahedra, Campi-Colesanti-Gronchi problems on minimal volume, the translative constant volume property, and Petty's polar projection problem, and it asserts that the latter two are equivalent. The survey is organized around the theme that some classical and recent volume functions do or do not determine a convex body.","tokens_in":13193,"tokens_out":6223,"duration_ms":70612,"significance":"If the claims are taken as established, the survey would be a useful map of old and recent problems in geometric tomography and volume-function characterization, with a clean identity (Lemma 1) connecting the convex-hull function to the brightness function and volume. The proof of Lemma 1 is correct and easy to check, and Theorem 5 gives an explicit construction principle that is plausible and checkable by hand. The literature summary broadly matches standard results in convex geometry, and the survey cites many relevant sources. However, the novel angle of the survey hinges on an unproved announced equivalence between the translative constant volume property and Petty's polar projection problem; as written, that equivalence is neither proved nor cited, so the advertised unification is currently unsupported. The planar example used to build the 3-dimensional counterexample is also under-specified. These issues are local and reparable, so the manuscript could be acceptable after a substantial revision that either proves or properly labels the equivalence and tightens the example.","major_comments":[{"comment":"The sentence 'The reason that we mentioned here the polar projection problem that it is equivalent to the conjecture on translative constant volume property. (We will prove it in a forthcoming paper.)' is an unsupported asserted equivalence, and it is load-bearing for the survey's stated novelty: it is what makes the translative constant volume conjecture a modern form of Petty's old problem. No proof, sketch, or reference is supplied. Since the equivalence appears to be short (using Eq. (12), the identity h_{ΠK} = b_K, and the polar formula ρ_{K^\\circ}(u) = 1/h_K(u), the translative constant volume property reduces to b_K(u)ρ_K(u) being constant, while ΠK^\\circ = λK reduces to the same relation), the author should either include this proof in the manuscript or explicitly label the equivalence as a conjecture/open problem. As it stands, the reader cannot distinguish an established theorem from a research announcement.","section":"§3, immediately after Conjecture 6"},{"comment":"The construction of the non-congruent nine-sided polygons P and Q is described only qualitatively ('add in a suitable manner ... three congruent suitable equilateral triangles') and relies on Figure 9. This example is load-bearing: Theorem 5 uses it to produce, in every dimension, non-congruent convex bodies with equal brightness and equal volume, and therefore to show that the convex-hull function does not determine a body. The text should either give an explicit coordinate construction, specify the required size and placement of the three triangles, or cite a reference where such a pair is constructed. The present level of detail does not allow the reader to verify convexity, equality of area, or equality of width.","section":"§3, planar pair construction (pp. 12–13)"}],"minor_comments":[{"comment":"The spelling 'covariagram' is used repeatedly; the standard term is 'covariogram'.","section":"Throughout"},{"comment":"The phrase 'as the author [4] proved' is misleading: reference [4] is Bianchi's paper, not a paper by the present author. Please write 'as Bianchi [4] proved' or 'as proved in [4]'.","section":"§2, paragraph on the covariogram problem"},{"comment":"The Gaussian curvature is denoted by K, which collides with the notation for a convex body K; using κ or another symbol would remove avoidable confusion.","section":"§2.3, differential equation for Blaschke body"},{"comment":"In the text this reference is cited as 'Bernd and Weber', but the authors are Kawohl and Weber. Please correct the citation.","section":"Reference [24]"},{"comment":"The sentence 'Since the covariogram function also holds this two properties in dimension n for n≥4 the answer should be \"no\"' is informal; the intended logic is that the covariogram determines volume and brightness and fails to determine K in dimension at least 4. Please spell out this implication explicitly.","section":"§3, Remark after Lemma 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a survey, and its main gate should be the correctness of the new equivalence claim and the explicit construction behind the counterexample. The unproved equivalence between the translative constant volume property and Petty's polar projection problem should be resolved before publication, either by including a proof (which the stress-test note suggests is short) or by clearly marking it as conjectural. The planar pair construction also needs to be made rigorous or referenced. The numerous typos and one misattribution suggest a careful proofreading pass is needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a survey, not a research paper, and the honest thing to say up front is that its main value is organizational. The genuinely new pieces are Lemma 1 — the convex-hull function equals volume plus |alpha| times brightness — and Theorem 5, which constructs non-congruent prisms with equal brightness and volume. Both are correct. Lemma 1 is a clean observation that connects the convex-hull function to the brightness function, and Theorem 5's proof is checkable and does what it claims. The survey also gives a decent map of the surrounding landscape: covariogram, width, brightness, the minimum-volume conjectures of Bonnesen–Fenchel, Heil, and Campi–Colesanti–Gronchi, and the translative constant-volume property. Someone new to this corner of convex geometry could use it to get oriented.\n\nThe soft spots are real. The biggest is the announced equivalence between Conjecture 5 (translative constant-volume property implies ellipsoid) and Petty's polar projection problem. Section 3 simply says the two are equivalent with a parenthetical that a proof will appear in a forthcoming paper. No sketch, no reference — nothing for a reader to verify. Your stress-test note shows the equivalence is likely true and the derivation is short, but as written it is load-bearing without support. That is not acceptable in a survey that advertises this as its novel angle. Either prove it or mark it as a conjecture.\n\nThe planar example in the remark after Lemma 1 is under-specified: the construction with two hexagons and three triangles needs a clearer description or a better figure. The statement of Alexandrov's projection theorem in the introduction is garbled — the version with V_i and arbitrary k-dimensional subspaces is not the standard theorem, and the general statement is likely false as written. There are also many language errors ('brigthness', 'ewerywhere', 'covariagram') that a referee would want cleaned up before publication.\n\nNone of this is fatal. The core mathematics that is actually proved is sound, the literature summaries are broadly correct, and the self-citations to the author's own earlier papers are legitimate uses of published results, not circular reasoning. The survey deserves a serious referee: the announced equivalence should be either removed, proved, or turned into an explicit conjecture, and the minor errors should be fixed. If the author follows through on the forthcoming proof, the survey becomes more interesting; as it stands, it is still a useful reference for specialists.\n\nRecommendation: send to peer review, with the expectation of moderate revision.","headline":"A useful survey of volume-function characterizations with a correct elementary lemma and prism construction, but the advertised equivalence with Petty's polar projection problem is asserted without proof and should be either proved or explicitly labeled a conjecture.","tokens_in":13787,"tokens_out":6968,"would_cite":true,"duration_ms":72794,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A40","52A38","26B15","52B11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The convex-hull function of a body is just its volume plus its shadow area.","keywords":["convex body","convex-hull function","covariogram function","brightness function","projection body","translative constant volume property","Radon curve","polar projection problem"],"falsifier":"Find a centrally symmetric convex body in $\\mathbb{R}^3$ that satisfies the translative constant-volume property, meaning the hull volume for touching translates is constant, but that is not an ellipsoid; such a body would disprove Conjecture 5, and under the asserted equivalence would also refute the polar projection conjecture.","tokens_in":12794,"feed_emoji":"📐","tokens_out":9864,"duration_ms":89486,"temperature":0.7,"pith_summary":"This survey gathers the classical volume-type functions used to characterize convex bodies—covariogram, width, brightness, and convex-hull functions—and asks which of them determine the body. Its central technical point is a short identity: when a body is translated by a vector along a unit direction, the volume of the convex hull of the body and its translate equals the body's volume plus the translation length times the area of the body's shadow in the perpendicular hyperplane. Consequently the convex-hull function contains no information beyond volume and brightness, so every pair of non-congruent bodies with equal volume and equal brightness automatically defeats it. The survey also connects the translative constant-volume property to Radon curves and to a long-standing polar projection problem, and it collects the still-open minimum-volume problems for constant width, constant thickness, and constant brightness. A careful reader comes away with a map of old conjectures that are now concrete enough for computational testing.","feed_headline":"The convex-hull function is volume plus shadow area","feed_subtitle":"A sliding body's hull volume adds only shadow area, so it cannot separate bodies that volume and shadows cannot.","key_machinery":"The load-bearing object is the convex-hull function $G_K(t)=\\operatorname{vol}(\\operatorname{conv}(K\\cup(K+t)))$, together with the identity $G_K(\\alpha u)=\\operatorname{vol}_n(K)+|\\alpha|\\operatorname{vol}_{n-1}(K|u^\\perp)$, where the second term is the brightness function, the area of the orthogonal projection of $K$ onto a hyperplane perpendicular to $u$. This identity, obtained from the Cavalieri principle, reduces every question about the convex-hull function to the classical pair (volume, brightness) and ties the whole survey together.","core_discovery":"The paper's central discovery is Lemma 1: for any convex body, unit direction, and real translation length, $G_K(\\alpha u)=\\operatorname{vol}_n(K)+|\\alpha|\\operatorname{vol}_{n-1}(K|u^\\perp)$; that is, the convex-hull function is determined by the volume and the brightness function. From this it follows immediately that the convex-hull function cannot characterize a convex body in any situation where volume and brightness fail to do so, and the survey exhibits such pairs via prisms built over equal-volume, equal-width polygons. Around this reduction, the survey organizes a century of related problems: the covariogram determines centrally symmetric bodies, plane bodies, and three-dimensional polytopes, but fails in dimension four and higher; bodies of constant width and constant brightness in three dimensions are balls; and the translative constant-volume property in the plane is equivalent to having a Radon curve as the boundary of the central symmetral. The open conjectures assert that in dimension three or higher this property forces an ellipsoid, and that the same conclusion holds for the polar projection problem.","pith_inferences":["If the asserted equivalence between the translative constant-volume property and the polar projection problem holds, then a computational search for non-ellipsoidal bodies with constant hull volume along touching translates would simultaneously test a seventy-year-old conjecture.","The reduction to brightness suggests that counterexample searches for the convex-hull function can be restricted to pairs of bodies with equal volume and equal brightness, a much smaller search space than arbitrary pairs of non-congruent bodies.","The paper's framing implies that the old minimum-volume problems are ripe for numerical optimization; the conjectured optimizers are all built from tetrahedra, so testing requires only few parameters.","Differentiating the convex-hull identity suggests that higher derivatives of $G_K$ may encode curvature information beyond the brightness function, potentially yielding finer invariants that separate bodies brightness cannot."],"forward_implications":["In every dimension, any two convex bodies with the same volume and the same brightness function have the same convex-hull function.","The convex-hull function therefore fails to determine a body in dimension four and above, where covariogram counterexamples already provide equal-volume, equal-brightness pairs.","In the plane, a convex body satisfies the translative constant-volume property exactly when the boundary of its central symmetral is a Radon curve, equivalently when the body has constant width in a Radon norm.","If the conjectures are true, the only centrally symmetric bodies in dimensions three and higher whose touching-translate hull volume is constant are ellipsoids.","Three-dimensional bodies of constant width and constant brightness are balls, and the minimum-volume candidates in the constant-width, constant-thickness, and constant-brightness classes are all tetrahedron-derived bodies."],"supporting_citations":[{"why":"Supplies the geometric-tomography background: Cavalieri principle, projection-body theorems, and the brightness surface-area measure equivalence.","marker":"[13]"},{"why":"Introduced the covariogram and posed the determination problem that anchors the survey's volume-function questions.","marker":"[29]"},{"why":"Provides the counterexamples showing the covariogram does not determine convex bodies in dimensions four and higher.","marker":"[4]"},{"why":"Establishes that generic simplicial polytopes are determined by their projection functions, a positive contrast to the counterexamples.","marker":"[14]"},{"why":"Proves the covariogram determines three-dimensional convex polytopes, marking the frontier of the open 3D case.","marker":"[5]"},{"why":"Gives the plane equivalence between the translative constant-volume property and Radon curves, the basis for the higher-dimensional conjecture.","marker":"[16]"},{"why":"Supplies the ellipsoid conclusion when a covariogram depends only on the Minkowski norm, the model for Conjecture 5.","marker":"[32]"},{"why":"Posed the polar projection problem that the survey claims is equivalent to the translative constant-volume conjecture.","marker":"[36]"},{"why":"Originated the convex-hull function and proved the convexity-of-hull-volume fact that motivates its study.","marker":"[12]"},{"why":"Provides the homothetic convex-hull results showing that norm-dependence forces balls or homothetic bodies, though the proofs do not cover the touching-translate case.","marker":"[9]"}],"fun_headline_variants":["Hull volume is just volume plus shadows","One formula ties hull volume to shadows","Volume and shadows decide hull volume—and fail to decide the body","A century of volume functions, one simple reduction","Convex hull volume = body volume + shadow area"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the translative constant-volume property is equivalent to the polar projection problem, an equivalence the paper states without proof and defers to a forthcoming paper.","fun_headline_variants_meta":{"raw":{"variants":["Hull volume is just volume plus shadows","One formula ties hull volume to shadows","Volume and shadows decide hull volume—and fail to decide the body","A century of volume functions, one simple reduction","Convex hull volume = body volume + shadow area"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000773,"raw_usage":{"total_tokens":3369,"prompt_tokens":840,"completion_tokens":2529,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":2456}},"tokens_in":456,"tokens_out":2529,"duration_ms":18110,"temperature":1.0,"reasoning_tokens":2456,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:21:09.348434+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a centrally symmetric convex body in $\\mathbb{R}^3$ that satisfies the translative constant-volume property, meaning the hull volume for touching translates is constant, but that is not an ellipsoid; such a body would disprove Conjecture 5, and under the asserted equivalence would also refute the polar projection conjecture.","supporting_citations":[{"cited_title":"J., Geometric Tomography Cambridge University Press, 1995","cited_arxiv_id":null,"evidence_quote":"Supplies the geometric-tomography background: Cavalieri principle, projection-body theorems, and the brightness surface-area measure equivalence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the covariogram and posed the determination problem that anchors the survey's volume-function questions."},{"cited_title":"London Math","cited_arxiv_id":null,"evidence_quote":"Provides the counterexamples showing the covariogram does not determine convex bodies in dimensions four and higher."},{"cited_title":"London Math","cited_arxiv_id":null,"evidence_quote":"Establishes that generic simplicial polytopes are determined by their projection functions, a positive contrast to the counterexamples."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the covariogram determines three-dimensional convex polytopes, marking the frontier of the open 3D case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the plane equivalence between the translative constant-volume property and Radon curves, the basis for the higher-dimensional conjecture."},{"cited_title":"Mathematika 40 (1993), 278–289","cited_arxiv_id":null,"evidence_quote":"Supplies the ellipsoid conclusion when a covariogram depends only on the Minkowski norm, the model for Conjecture 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Posed the polar projection problem that the survey claims is equivalent to the translative constant-volume conjecture."},{"cited_title":"Der zentralsymmetrische Kern und die zentralsymmetrische H¨ ulle von konvexen K¨ orpern.Math","cited_arxiv_id":null,"evidence_quote":"Originated the convex-hull function and proved the convexity-of-hull-volume fact that motivates its study."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the homothetic convex-hull results showing that norm-dependence forces balls or homothetic bodies, though the proofs do not cover the touching-translate case."}],"review_version":1}