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Floating bodies for ball-convex bodies

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Pith's one-line read The paper proves that the volume lost by cutting an R-ball-convex body with R-balls has a universal δ^{2/(n+1)} limit given by an integral of (curvature − 1/R)^{1/(n+1)}, defining a relative affine surface area.

desk verdict Solid extension of classical floating body theory to ball-convex bodies; the main theorem is likely correct, but the lower-bound proof in Lemma 8 needs a clean rewrite before I'd fully trust it. read the letter →

arxiv 2504.15488 v1 pith:XIVPN5E7 submitted 2025-04-21 math.MG

Carsten Schuett , Elisabeth M Werner , Diliya Yalikun This is my paper · ORCID
classification math.MG MSC 52A2052A3853A15
keywords ball-convexbodiesfloatingbodyaffinesurfacearearelativevaluationuppersemicontinuityconvexgeometrycurvature
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

This paper extends the classical floating-body construction from convex bodies to ball-convex bodies: bodies that can be written as intersections of congruent R-balls. It defines the R-ball floating body by cutting off caps with R-balls, and proves a sharp asymptotic formula for the volume decrease as the cap volume goes to zero. The coefficient is an integral over the boundary of the product of $(\kappa_i - 1/R)^{1/(n+1)}$, which the authors name the relative affine surface area. A sympathetic reader should care because this gives ball-convex geometry its own affine surface area with the same formal properties as the classical one: rigid-motion invariance, valuation, homogeneity, and upper semicontinuity, with the classical affine surface area recovered as $R\to\infty$.

What carries the argument

The central object is the R-ball floating body $K_R^\delta = \bigcap_{\operatorname{vol}_n(K\setminus (z+RB_2^n))\le \delta} (z+RB_2^n)$, the intersection of all R-balls that cut off at most $\delta$ of K's volume. The proof machinery combines three ingredients: Lemma 7 compares the volume of the cap removed from K with caps of the approximating ellipsoids $E(\varepsilon^-)$ and $E(\varepsilon^+)$ that squeeze $\partial K$ near a boundary point (equation (30), imported from [46]); Lemma 8 converts this cap-volume comparison into the pointwise limit of $\|x-x_\delta\|/\delta^{2/(n+1)}$; and Proposition 2 evaluates the resulting spherical integral as a product of square roots, yielding the explicit constant $c_n$. Lemma 6, based on McMullen's rolling function $r_K(x)$, controls the convergence uniformly over the boundary so that integration and limit can be interchanged.

What would settle it

For a planar ellipse that is R-ball convex with both principal curvatures strictly greater than $1/R$, compute the exact area of $K_R^\delta$ by direct integration for small $\delta$ and compare the coefficient of $\delta^{2/3}$ with $c_2\int_{\partial K}(\kappa-1/R)^{1/3}\,ds$, where $c_2=\frac12(3/2)^{2/3}$; any disagreement in the leading coefficient would falsify Theorem 1.

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Extended reading notes

Core claim

The central discovery is Theorem 1: for every R-ball convex body K whose principal curvatures are all strictly larger than $1/R$, the volume difference between K and its R-ball floating body satisfies $$\lim_{\delta\to 0} \frac{\operatorname{vol}_n(K)-\operatorname{vol}_n(K_R^\delta)}{\$delta^{{2/(n+1)}}$} = c_n \int_{\partial K} \prod_{i=1}^{n-1} \left(\kappa_i(K,x)-\frac{1}{R}\right)^{1/(n+1)} d\mu_K(x),$$ with the explicit constant $c_n = \frac{1}{2}\left(\frac{n+1}{\operatorname{vol}_{n-1}(B_2^{n-1})}\right)^{2/(n+1)}$. This justifies calling the integral the relative affine surface area $\operatorname{as}_R(K)$, and it recovers Blaschke's affine surface area when $R\to\infty$.

Load-bearing premise

The formula rests on the assumption that near each boundary point the body can be squeezed between two nearly identical ellipsoids in a neighborhood large enough to contain the caps that the cutting R-balls remove; if that ellipsoidal approximation is not uniform along the boundary, the cap-volume comparison that produces the limit breaks down.

Editorial extensions

If this is right

  • The formula defines a rigid-motion invariant and upper-semicontinuous valuation on R-ball-convex bodies, giving ball-convex geometry a natural affine-analytic invariant alongside its metric ones.
  • Taking $R\to\infty$ recovers the classical affine surface area, so the new notion is a genuine one-parameter relative version rather than a separate construction.
  • The inequality $\operatorname{as}_R(K) \le n\operatorname{vol}_n(B_2^n)^{2/(n+1)} \operatorname{vol}_n(K)^{(n-1)/(n+1)}$, with equality only for $R=\infty$ and ellipsoids, provides a relative affine isoperimetric bound.
  • R-ball polyhedra, intersections of finitely many R-balls, have $\operatorname{as}_R=0$, which is consistent with the upper semicontinuity and with the fact that every smooth body can be approximated by such polyhedra.
  • In dimension 2 the construction recovers the r-spindle floating body, linking the result to existing approximation questions for random disc polygons.

Reading between the lines

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

  • A natural extension the authors flag is an $L_p$-version of $\operatorname{as}_R$; if the derivative formula holds with a $p$-power weight, it would yield a family of relative affine invariants on ball-convex bodies.
  • The constant $c_n$ being exactly the classical floating-body constant suggests the relative formula is the leading term of an expansion in $1/R$, possibly connecting to spherical or hyperbolic floating bodies when the ball radius is allowed to become imaginary.
  • One could probe stability by computing the second-order term in $\delta$ for small $n$; the theorem fixes only the leading order, and the next coefficient would distinguish genuinely different relative affine structures.
  • The strict curvature assumption $\kappa_i>1/R$ suggests that the boundary case $\kappa_i=1/R$, where the integrand vanishes, may produce a different power of $\delta$; testing this on bodies with flat arcs made of R-ball pieces would clarify the boundary behavior of the relative affine surface area.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper introduces a floating-body construction for ball-convex bodies. For an R-ball convex body K, the R-ball floating body K_R^δ is defined as the intersection of all R-balls that cut off from K a set of volume at most δ. The main result (Theorem 1) asserts that, if all principal curvatures of K are strictly larger than 1/R, then the right derivative of the volume difference at δ=0 has the explicit limit c_n ∫_{∂K} ∏_{i=1}^{n-1} (κ_i(K,x) − 1/R)^{1/(n+1)} dμ_K(x). The integral, called the relative affine surface area as_R(K), recovers the classical affine surface area as R→∞. The paper proves that as_R is invariant under rigid motions, homogeneous, a valuation, and upper semicontinuous, and it establishes an affine isoperimetric-type inequality. The proofs combine a cap-volume estimate for ellipsoidal caps cut by R-balls (Lemma 7), a pointwise limit for the radial displacement x−xδ (Lemma 8), and an integration argument using a bound from the rolling function (Lemma 6).

Significance. If Theorem 1 is correct, the paper provides a natural extension of affine surface area to ball-convex bodies, with a geometric definition via floating bodies and the expected R→∞ limit. The proof structure is largely self-contained: Proposition 2 is proved from first principles, Lemma 7 supplies two-sided cap estimates with explicit constants, and Lemma 6 gives the integrable control needed for dominated convergence. The additional valuation and semicontinuity properties make the new functional a promising tool for approximation theory of ball-convex bodies, where only planar versions were previously available. The main weakness is that the pointwise lower bound in Lemma 8, and the uniformity of the ellipsoidal approximation used there, are not fully established; these gaps are localized and appear fixable, but they are load-bearing for the central limit formula.

major comments (2)
  1. [§3.2, Lemma 8, inequality (32)] The lower-bound part of Lemma 8, which is the only source of the lower bound in Theorem 1, is not rigorously established. The two-sided estimate (32) as displayed is not dimensionally homogeneous, so it cannot be checked as written. The derivation of the left-hand side contains an invalid step: after obtaining an inequality with the extra term 2(xδ(2)−a_n)a_n ξ(1)^3/((1−ε)a1)^4, the proof drops this term and concludes the inequality without it, even though the assumption a_n ≥ xδ(2) makes the dropped term non-positive, which weakens the right-hand side rather than preserving the inequality. In addition, the paragraph after (32) asserts that the R-ball centered at (R+d0)e_n 'cuts off strictly less than δ from K as xδ is in the interior of this R-ball' without a proof; this is not a consequence of interiority alone and needs a quantitative argument relating vol(K\B) to δ. Please replace (32) and the subsequent paragraph by a complete, dimensionally consistent derivation.
  2. [§3.2, equations (30)–(31)] The ellipsoidal approximation (30) is quoted from [46] for a fixed boundary point x, but Lemma 8 requires a uniform version along the sequence xδ→x: the proof asserts that for all sufficiently small δ and all support R-balls at xδ, the set E(ε−)\(z+RB^n) is contained in the fixed neighborhood H^-(x−Δε e_n,e_n)∩E(ε−). This uniformity is not proved and does not follow from the pointwise statement (30), because the support ball may vary with δ. Since the cap-volume comparison of Lemma 7 is transferred from K to E(ε±) through this inclusion, the lower bound in Lemma 8 depends on this missing uniformity. Please provide a proof or a precise citation for the uniform version.
minor comments (6)
  1. [§3.2, proof of Lemma 6] The sentence 'Such an R-ball exists by Theorem 5 (i)' should refer to Lemma 5; the paper contains no Theorem 5.
  2. [§3.2, proof of Lemma 6] Equation (20) contains a corrupted expression: 'B^n_2(x−r_K(x)N_K(x), r_K(x)−K(x))' should presumably be 'B^n_2(x−r_K(x)N_K(x), r_K(x))'.
  3. [§3.1, Proposition 2] The statement of Proposition 2 is for integrals over S^{n−1}, while the proof in §3.1 treats the S^{n−2} case used later; please make the reduction explicit.
  4. [§3.3, Proposition 3(iv)] The application of [32] is terse; please spell out the hypotheses of the cited semicontinuity theorem and verify them for the integrand f(∏(κ_i−1/R)) on the class of R-ball convex bodies.
  5. [§2, Definition 2] In the definition of as_L(K), the term κ_i(L, N_L^{-1}(N_K(x))) is undefined when N_K(x) is not a regular value of the Gauss map of L; please add a convention for such boundary points.
  6. [Introduction and affiliations] There are several typos: 'surface surface area' in the Introduction, the duplicated email address in the author affiliation, and the inconsistent spelling 'covarigram' for 'covariogram' in §2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 derives the floating-body limit from first principles and local ellipsoidal approximation; the relative affine surface area is defined after the theorem, not used as its input.

full rationale

The paper's central result, Theorem 1, computes lim_{δ→0} (vol_n(K)-vol_n(K_R^δ))/δ^{2/(n+1)} and identifies the limit as an explicit integral of (κ_i(K,x)-1/R)^{1/(n+1)}. The quantity as_R(K) is introduced in Definition 2 only after the theorem, so the equality between the floating-body derivative and the integral is the content of the theorem rather than an input. The proof proceeds through Lemmas 4–8: volume comparison (Lemma 4), existence of support R-balls (Lemma 5), a bound on cap depths (Lemma 6), an explicit cap-volume estimate for ellipsoids (Lemma 7), and the pointwise limit (Lemma 8). Lemma 8 uses the standard local ellipsoidal approximation of a smooth convex body near a boundary point, quoted from [46], and Lemma 6 follows the template of Lemma 6 of [45]. These are prior works with overlapping authors, but the cited results are parameter-free geometric approximation statements whose assumptions do not include the target floating-body limit or the definition of relative affine surface area. In particular, equation (30) is a standard second-order boundary approximation and is not equivalent to Theorem 1. There is no fitted constant renamed as a prediction, no uniqueness theorem imported to force the ansatz, and no definition that presupposes the limit formula. The skeptical concern about the derivation of inequality (32) in Lemma 8 concerns a possible gap or unproven estimate in the proof, not circularity; even if that estimate were incomplete, it would not make the theorem an input to itself. Accordingly, no circular step is present, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

No parameters are fitted; R is part of the input class and c_n is an explicit constant. The proofs rely on standard convex geometry results and on prior work by the same authors for technique, which is legitimate support but raises the burden on careful checking of the cited lemmas.

assumptions (6)
  • domain assumption Second-order differentiability of convex boundaries a.e. and existence of generalized principal curvatures (Alexandrov, Busemann-Feller)
    Used throughout Section 2 to define κ_i(K,x), the Gauss map, and the integrand in Theorem 1.
  • standard math Volume difference formula in Lemma 4 from [45]
    Entry point of the proof of Theorem 1, expressing vol(K)-vol(K_R^δ) as a boundary integral over ∂K.
  • standard math Integrability bound (12) for powers of the rolling radius, from McMullen [36] and [45]
    Needed to justify dominated convergence when passing the limit under the boundary integral in Theorem 1.
  • domain assumption Local ellipsoidal approximation of ∂K by E(ε−) and E(ε+) as in equation (30), quoted from [46]
    Core comparison in Lemma 8; its validity for R-ball convex bodies is imported, not proven here.
  • standard math Ludwig's upper semicontinuity theorem for curvature integrals [32]
    Used in Proposition 3(iv); hypotheses are not rechecked in the text.
  • standard math Classical affine isoperimetric inequality for affine surface area
    Used in Proposition 3(iii) through the inequality as_R(K) ≤ as(K).
invented entities (1)
  • relative affine surface area as_R(K) and as_L(K)
    purpose: New boundary integral claimed to equal the floating-body volume derivative limit and to satisfy valuation, homogeneity, and upper semicontinuity.
    A newly defined mathematical quantity rather than a physical entity; the paper's theorems are the only evidence for its status, and no external falsifiable measurement exists.

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Pith. "Pith review of Floating bodies for ball-convex bodies." pith.science (2026). https://pith.science/paper/XIVPN5E7

@misc{pith2026250415488,
  author       = {Pith},
  title        = {Pith review of: Floating bodies for ball-convex bodies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XIVPN5E7}},
  note         = {Machine review of arXiv:2504.15488}
}
abstract

We define floating bodies in the class of $n$-dimensional ball-convex bodies. A right derivative of volume of these floating bodies leads to a surface area measure for ball-convex bodies which we call relative affine surface area. We show that this quantity is a rigid motion invariant, upper semi continuous valuation.

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Figure 1
Figure 1. The angles the sine law for the triangle ∆ = sin β sin γ ∥x − xδ∥. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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Cited by 1 Pith paper

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

  1. Some inequalities of isoperimetric type for the c-affine surface area

    math.MG 2025-05 conditional novelty 6.0 of 10

    For ball-bodies, the c-affine surface area is maximized by the ball of radius n/(n+1), and the product with its c-dual is bounded by the squared value at the ball of radius 1/2.

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