Pith. sign in

REVIEW 5 minor 23 references

Anisotropic area measures of convex bodies

T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read If a convex body's k-th anisotropic area measure is proportional to its top one, the body is a k-tangential body of the gauge body.

desk verdict A clean, standard-machinery extension of the ball case to arbitrary regular strictly convex gauge bodies; the new measures and characterization are correct and worth refereeing. read the letter →

arxiv 2506.08803 v3 pith:3BL7CRKW submitted 2025-06-10 math.MG

Rolf Schneider This is my paper
classification math.MG MSC 52A2053C4253B25
keywords anisotropicsupportmeasurearearelativedifferentialgeometrytangentialbodymixedconvexAlexandrov-Fenchelinequalitygauge
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

Motivated by relative differential geometry, this paper replaces the unit ball by a gauge body $E$ and defines anisotropic area measures $S^E_k(K,\cdot)$ for convex bodies $K$. Its main theorem states that, for $k\in\{0,\dots,n-2\}$, the relation $S^E_k(K,\cdot)=cS^E_{n-1}(K,\cdot)$ holds with a constant $c$ if and only if $K$ is a translated and scaled copy of a $k$-tangential body of $E$. This generalizes the earlier Euclidean result that proportional area measures characterize $k$-tangential bodies of a ball, and it gives a measure-theoretic route to tangential-body rigidity without differentiability assumptions. Along the way the paper introduces anisotropic support measures, which specialize to both the new anisotropic area measures and anisotropic curvature measures, and shows that the top-order anisotropic area measure reduces the relative Minkowski problem to the classical one.

What carries the argument

The central object is the anisotropic area measure, defined by $S^E_k(K,\alpha)=S(K[k],E[n-1-k],u_E(\alpha))$, where $S$ is the mixed area measure and $u_E$ maps each boundary point of $E$ to its unique outer unit normal vector. This measure is concentrated on the boundary of $E$ and, for sufficiently smooth bodies, equals the integral over $\alpha\cap\partial E$ of the $k$-th normalized elementary symmetric function of the relative principal radii. The proof mechanism is the mixed-volume expansion: the $(n-1)$-dimensional Hausdorff measure of the boundary of the parallel body $K+\rho E$ expands in powers of $\rho$ with coefficients built from anisotropic support measures, and integrating the assumed measure equality turns it into equalities among mixed volumes. Those equalities feed the Alexandrov-Fenchel inequality chain, whose equality case (Bol's theorem) is the structural step producing the tangential-body conclusion.

What would settle it

Compute the anisotropic area measures for a sufficiently smooth convex body $K$ and a smooth strictly convex gauge body $E$ such that $K$ is not homothetic to a $k$-tangential body of $E$; using $S^E_k(K,\alpha)=\int_{\alpha\cap\partial E}s^E_k\,dH^{n-1}$, check whether $s^E_k/s^E_{n-1}$ is constant on $\partial E$. If it is constant, the theorem fails.

Watch

Extended reading notes

Core claim

The paper's central claim is the characterization in Theorem 1. For a fixed gauge body $E$ (regular, strictly convex, containing the origin in its interior), a convex body $K$ satisfies $S^E_k(K,\alpha)=c\,S^E_{n-1}(K,\alpha)$ for some constant $c$ and all Borel sets $\alpha$, with $k\in\{0,\dots,n-2\}$, exactly when $K$ is homothetic to (a translated and scaled copy of) a $k$-tangential body of $E$. A $k$-tangential body of $E$ is a convex body that contains a homothetic copy of $E$ and has the property that every supporting hyperplane not supporting that copy contains only $(k-1)$-singular boundary points. The proof first replaces $K$ by a homothet so that $c=1$, integrates the measure equality against support functions to obtain the mixed-volume equalities $V^E_{k+1}=V^E_n$ and $V^E_k=V^E_{n-1}$, and then runs the Alexandrov-Fenchel inequality chain to force equality everywhere. Bol's theorem converts the resulting equality into the fact that $K$ is homothetic to an $(n-2)$-tangential body of $E$, and Favard's criterion upgrades this to a $k$-tangential body. The converse direction starts from Favard's criterion and uses a proportionality theorem for mixed area measures to recover the measure identity.

Load-bearing premise

The proof depends on an external equality-case classification (Bol's theorem) that turns one mixed-volume equality into the assertion that K is homothetic to an (n-2)-tangential body of E, and the paper does not reproduce or verify the hypotheses of that classification beyond its standing assumptions.

Editorial extensions

If this is right

  • For any gauge body $E$, the proportionality of $S^E_k$ and $S^E_{n-1}$ is a complete characterization of $k$-tangential bodies of $E$, so a single measure comparison reveals this geometric structure.
  • Setting $E$ to the unit ball recovers the classical result: area measures of orders $k$ and $n-1$ are proportional exactly for $k$-tangential bodies of a ball.
  • The top-order anisotropic area measure $S^E_{n-1}(K,\cdot)$ is the pullback of the Euclidean surface area measure under the Gauss map of $E$, so the relative Minkowski problem has the same solvability conditions as the classical one.
  • The anisotropic support measures unify area and curvature measures as boundary coefficients of a parallel-body expansion, giving a common technical language for both.
  • The proof shows that the chain of mixed-volume equalities $V^E_n=V^E_{n-1}=\cdots=V^E_k$ is equivalent to $K$ being a $k$-tangential body of $E$, so the measure proportionality and this volume chain are interchangeable conditions.

Reading between the lines

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

  • A stability version is a natural next step: one would expect the deviation of $S^E_k(K,\cdot)$ from being proportional to $S^E_{n-1}(K,\cdot)$ to control how far $K$ is from a $k$-tangential body of $E$, but the paper does not quantify this.
  • The regularity assumptions on $E$ could likely be relaxed by approximation; if the characterization is stable under limits, the same theorem should hold for general gauge bodies, though proving that would require additional arguments.
  • In smooth relative differential geometry, the theorem says that if the relative radius symmetric function $s^E_k$ is a constant multiple of $s^E_{n-1}$ on the boundary of $E$, the hypersurface must be a tangential body; this connects the measure result to rigidity statements for anisotropic curvature functions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper introduces, for a fixed regular strictly convex gauge body E containing the origin, anisotropic area measures S^E_k(K,·) and more generally anisotropic support measures for a convex body K, defined through mixed area measures and the inverse Gauss map of E. In the smooth case these measures integrate normalized elementary symmetric functions of relative radii. The main theorem (Theorem 1) states that for k=0,...,n-2, S^E_k(K,·) is proportional to S^E_{n-1}(K,·) if and only if K is homothetic to a k-tangential body of E. The proof uses the Alexandrov-Fenchel inequalities, Bol's equality case, Favard's characterization of tangential bodies, and Theorem 7.4.6 from the author's book.

Significance. The theorem generalizes the Euclidean result for balls to an arbitrary gauge body and gives a sharp, measure-theoretic counterpart to relative differential geometry. The new anisotropic support measures are a natural common generalization of anisotropic area and curvature measures, and the Minkowski problem remark is useful. The proof is concise and correct modulo standard external theorems; the main value lies in the formulation and synthesis.

minor comments (5)
  1. [§4, after eq. (16)] The proof invokes Bol's theorem [3] (reproduced as [20, Thm. 7.6.19]) to conclude from (V^E_{n-1})^2 = V^E_{n-2}V^E_n that K is homothetic to an (n-2)-tangential body of E; since this is the deepest step, please state the relevant equality case or at least spell out its hypotheses so readers can verify applicability.
  2. [§4, last paragraph] The application of [20, Thm. 7.4.6] is very terse: the definitions of V(i), S(i), and condition (b) are not reproduced. A sentence explaining how (18) implies condition (b) would improve self-containedness.
  3. [§4, proof of (17)] In the sentence after 'Since K is a tangential body of rE', the pointwise equality is written as (h_K ◦ u_K)(x) = (h_E ◦ u_K)(x); it should be = r (h_E ◦ u_K)(x) to be consistent with h_K(u) = r h_E(u) and with (17).
  4. [Introduction, first paragraph] The phrase 'a corresponding results' should be 'a corresponding result'.
  5. [Abstract] The abstract says 'the area measure of order n-1' where 'the anisotropic area measure of order n-1' is meant; please clarify to avoid ambiguity with the classical surface area measure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof reduces the measure condition to mixed-volume equalities and then invokes independent classical theorems (Bol, Favard, Alexandrov–Fenchel equality cases), with no fitted parameter or self-referential definition.

full rationale

The anisotropic area measures are defined independently of the conclusion: S^E_k(K, α) = S(K[k], E[n−1−k], u_E(α)), and the target notion (k-tangential body of E) is defined geometrically in [20, Sect. 2.2], not as the equality being proved. In the forward direction, the condition S^E_k = S^E_{n-1} is first normalized and then converted, via the mixed-volume identity (3), into the two equalities V^E_{k+1}=V^E_n and V^E_k=V^E_{n-1}. These force equality throughout the Alexandrov–Fenchel chain, so (V^E_{n-1})^2 = V^E_{n-2} V^E_n. The inference from this equality to 'K is homothetic to an (n−2)-tangential body of E' is precisely Bol's theorem, an external classical result cited to its original source [3] and reproduced as [20, Thm. 7.6.19]; it does not assume Theorem 1. The subsequent step showing V^E_n = r V^E_{n-1} is proved in the paper by a direct boundary integral. The final characterization of k-tangential bodies via equal mixed volumes is Favard's theorem [5], again external; the converse applies the standard equality-case theorem [20, Thm. 7.4.6] with the correspondence V(i)=V^E_{n-i}, S(i)=S^E_{n-1-i}(K,·). None of these inputs depends on the paper's own conclusion, and no parameter is fitted to data. The reliance on the author's textbook [20] for standard mixed-volume theorems is a self-citation, but the theorems are external, parameter-free, and originally due to other authors where relevant, so it does not make the derivation circular. The only residual risk—whether the hypotheses of Bol's theorem are met exactly as indexed—is a correctness or verification concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The theorem is proved from standard convex geometry results. There are no fitted constants, no invented physical entities, and no data. The only substantive external inputs are the deep equality-case theorems (Bol, Favard, and Theorem 7.4.6 from [20]).

assumptions (5)
  • standard math Alexandrov-Fenchel inequalities for mixed volumes: V(K[j], E[n-j]) is log-concave in j.
    Cited as [20, Thm. 7.3.1]; used in Section 4 to obtain the chain of ratios from V^E_{n-1}/V^E_n down to V^E_k/V^E_{k+1}.
  • standard math Bol's equality-case theorem: if (V^E_{n-1})^2 = V^E_{n-2} V^E_n, then K is homothetic to an (n-2)-tangential body of E.
    Cited as [3], reproduced in [20, Thm. 7.6.19]; the key step after deriving the mixed-volume equalities (16).
  • standard math Favard's theorem: for K containing a translate of E, K is a k-tangential body iff V^E_k = V^E_{k+1} (equiv. (7.149)).
    Cited as [5, p.273] and [20, Thm. 7.6.17]; used in both directions of Section 4.
  • standard math Schneider's Theorem 7.4.6 in [20]: if mixed volumes of a set of bodies satisfy the indicated equalities, the associated area measures are proportional.
    Applied in the converse direction with m=n-k, K_0=K, K_1=E, C=(K[k]); condition (b) satisfied via (18), conclusion (d) yields proportional S^E_{n-1} and S^E_k.
  • domain assumption The Gauss map u_E: bd E -> S^{n-1} is a homeomorphism, which requires E to be regular and strictly convex.
    Stated in Section 2; needed for Definition 1 and for the equivalence (15). This is an assumption on the gauge body, not proved in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anisotropic area measures of convex bodies." pith.science (2026). https://pith.science/paper/3BL7CRKW

@misc{pith2026250608803,
  author       = {Pith},
  title        = {Pith review of: Anisotropic area measures of convex bodies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BL7CRKW}},
  note         = {Machine review of arXiv:2506.08803}
}
abstract

Motivated by the relative differential geometry, where the Euclidean normal vector of hypersurfaces is generalized by a relative normalization, we introduce anisotropic area measures of convex bodies, constructed with respect to a gauge body. Together with the anisotropic curvature measures, they are special cases of the newly introduced anisotropic support measures. We show that a convex body in ${\mathbb R}^n$, for which the anisotropic area measure of some order $k\in\{0,\dots,n-2\}$ is proportional to the area measure of order $n-1$, must be a $k$-tangential body of the gauge body.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Aleksandrov, A.D., Uniqueness theorems for surfaces in the large. I. (in Russian) Vestnik Leningrad. Univ. 11 (1956), 5–17

    work page 1956

  2. [2]

    Anisotropic curvature measures and volume preserving flows

    Andrews, B., Lei, Y., Wei, Y., Xiong, C., Anisotropic curvature measures and volume preserving flows. arXiv:2108.02049v1

    work page Pith review arXiv

  3. [3]

    Minkowski

    Bol, G., Beweis einer Vermutung von H. Minkowski. Abh. Math. Sem. Univ. Ham- burg 15 (1943), 17–56

    work page 1943

  4. [4]

    Reprint: Chelsea Publ

    Bonnesen, T., Fenchel, W., Theorie der konvexen K¨ orper.Springer, Berlin, 1934. Reprint: Chelsea Publ. Co., New York, 1984. English translation: BCS Associates, Moscow, Idaho, 1987

    work page 1934

  5. [5]

    Favard, J., Sur les corps convexes. J. Math. Pures Appl. , S´ erie 9,12 (1933), 219– 282

    work page 1933

  6. [6]

    Acta Math

    He, Y.J., Li, H.Z., Integral formula of Minkowski type and new characterization of the Wulff shape. Acta Math. Sinica 24 (2008), 697–704. 10

    work page 2008

  7. [7]

    Indiana Univ

    He, Y., Li, H., Ma., H., Ge, J., Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures. Indiana Univ. Math. J. 58 (2009), 853– 868

    work page 2009

  8. [8]

    Hsiung, C.C., Some integral formulas for closed hypersurfaces. Math. Scand. 2 (1954), 286–294

    work page 1954

Show all 23 references
  1. [9]

    Kohlmann, P., Characterizations via linear combinations of curvature measures. Arch. Math. 70 (1998), 250–256

    1998

  2. [10]

    Liebmann, H., Beweis zweier S¨ atze ¨ uber die Bestimmung von Ovaloiden durch das Kr¨ ummungsmaß oder die mittlere Kr¨ ummung f¨ ur jede Normalenrichtung.Nachr. Ges. Wiss. G¨ ottingen(1899), 134–142

  3. [11]

    A., Compact hypersurfaces: The Aleksandrov theorem for higher order mean curvatures

    Montiel, S., Ros. A., Compact hypersurfaces: The Aleksandrov theorem for higher order mean curvatures. In: Differential geometry. A Symposium in Honor of Man- fredo do Carmo. (B. Lawson, K. Tenenblat, eds.) Pitman Monogr. Surveys Pure Appl. Math., 52. Longman Scientific & Tech...

    1991

  4. [12]

    Duke Math

    Reilly, R.C., The relative differential geometry of nonparametric hypersurfaces. Duke Math. J. 43 (1976), 705–721

    1976

  5. [13]

    In- diana Univ

    Reilly, R.C., Applications of the Hessian operator in a Riemannian manifold. In- diana Univ. Math. J. 26 (1977), 459–472

    1977

  6. [14]

    Ros, A., Compact hypersurfaces with constant higher order mean curvature. Rev. Mat. Iberoamericana 3 (1987), 447–453

    1987

  7. [15]

    Scheuer, J., Zhang, X., Stability of the Wulff shape with respect to anisotropic curvature functionals. J. Functional Anal. 288 (2025), Paper No. 110715, 27pp

    2025

  8. [16]

    Teubner, Leipzig, 1962

    Schirokow, P.A., Schirokow, A.P., Affine Differentialgeometrie. Teubner, Leipzig, 1962

    1962

  9. [17]

    Schneider, R., Zur affinen Differentialgeometrie im Großen. I. Math. Zeitschr. 101 (1967), 375–406

    1967

  10. [18]

    23 (1978), 269–278

    Schneider, R., ¨Uber Tangentialk¨ orper der Kugel.Manuscripta Math. 23 (1978), 269–278

    1978

  11. [19]

    Schneider, R., Bestimmung konvexer K¨ orper durch Kr¨ ummungsmaße.Comment. Math. Helvetici 54 (1979), 42–60

    1979

  12. [20]

    2nd edn., Encyclo- pedia of Mathematics and Its Applications, vol

    Schneider, R., Convex Bodies: The Brunn–Minkowski Theory. 2nd edn., Encyclo- pedia of Mathematics and Its Applications, vol. 151, Cambridge University Press, Cambridge, 2014

    2014

  13. [21]

    Simon, U., Minkowskische Integralformeln und ihre Anwendungen in der Differ- entialgeometrie im Großen. Math. Ann. 173 (1967), 307–321. 11

    1967

  14. [22]

    Zur relativen Differentialgeometrie V: ¨Uber Eihyperfl¨ achen im Rn+1

    S¨ uss, W. Zur relativen Differentialgeometrie V: ¨Uber Eihyperfl¨ achen im Rn+1. Tˆ ohoko Math. J.31 (1929), 202–209

    1929

  15. [23]

    Indiana Univ

    Xia, C., On an anisotropic Minkowski problem. Indiana Univ. Math. J. 62 (2013), 1399–1430. Author’s address: Rolf Schneider Mathematisches Institut, Albert–Ludwigs-Universit¨ at D-79104 Freiburg i. Br., Germany E-mail: rolf.schneider@math.uni-freiburg.de 12

    2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.