arxiv: 2604.01950 · v4 · submitted 2026-04-02 · 🧮 math.MG

Recognition: no theorem link

Self perimeter of convex sets

Gershon Wolansky

Pith reviewed 2026-05-13 20:46 UTC · model grok-4.3

classification 🧮 math.MG

keywords convex setsself-perimeternormed spacesvolume definitionaffine invariancepolar dualitysurface measureAlexandrov problem

0 comments

The pith

A volume for unit balls in any normed space is defined so perimeter over volume equals the dimension n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a self-volume for the unit ball of a normed space that keeps the classical ratio of self-perimeter to volume equal to n. This volume is constructed to stay invariant under origin-fixing affine transformations and under polar duality. In two dimensions an explicit integral formula is given for the self-perimeter and extended to convex sets that lack central symmetry. The same idea is lifted to higher dimensions by a recursive integration that reduces the n-dimensional volume to (n-1)-dimensional volumes of planar sections of the boundary. An inverse problem is then solved perturbatively, showing that generic first-order changes to the Euclidean surface measure produce convex bodies with four-fold rotational symmetry.

Core claim

A natural volume V is defined for the unit ball B in an n-dimensional normed space so that the self-perimeter P satisfies P(B)/V(B) = n. The definition is built to be invariant under origin-preserving affine maps and under polar duality. For n=2 the self-perimeter admits an explicit integral expression over the boundary that extends to non-symmetric sets. In higher dimensions the volume is obtained recursively by integrating the (n-1)-dimensional self-volumes of planar intersections with the boundary. The associated Alexandrov-type problem for the surface measure admits perturbative solutions in which any generic perturbation of the Euclidean disk produces a four-fold symmetric convex set to

What carries the argument

The self-volume obtained by recursive integration of (n-1)-dimensional volumes of planar boundary intersections.

If this is right

The self-volume is invariant under origin-preserving affine transformations.
The self-volume is invariant under polar duality.
In two dimensions the self-perimeter is given by an explicit integral formula.
Generic perturbations of the Euclidean surface measure yield four-fold symmetric convex sets at first order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction supplies a canonical volume functional on Minkowski spaces without requiring an external Lebesgue measure.
Explicit evaluation of the self-volume for the l1 or l_infty unit ball would provide a direct numerical test of the ratio-n property.
The four-fold symmetry result indicates that the linearised inverse problem has a kernel tied to order-four rotations.
Similar recursive volume definitions could be examined for other intrinsic functionals such as surface area in convex geometry.

Load-bearing premise

That a recursive integration of lower-dimensional boundary volumes produces a consistent, finite volume for every convex body.

What would settle it

Compute the self-perimeter integral for the unit ball of the l1 norm in R^2, compute the corresponding self-volume by the recursive rule, and check whether their ratio equals exactly 2.

read the original abstract

This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball $B$ in $R^n$. We show that this volume definition is invariant under origin-preserving affine transformations and polar duality. For $n=2$, we derive an explicit integral formula for the self-perimeter of the unit ball, extend it to non-centrally symmetric sets;. The construction is extended to $\mathbb{R}^n$ via a recursive integration over the boundary, utilizing $(n-1)$-dimensional volumes of planar intersections. Finally, we pose and discuss an Alexandrov-type problem for the associated surface measure, providing perturbative solutions in the 2D case. In particular we prove that, generically, any perturbation of the surface measure of the Euclidean 2-D disk yields a 4-fold symmetric convex set in the leading order.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Defines a volume for unit balls in normed spaces that forces P/V = n by construction, with affine and duality invariance plus an explicit 2D formula and perturbative symmetry result, but the recursive extension needs a consistency check.

read the letter

The main new piece is a volume definition for the unit ball in a general normed space that is built to keep the perimeter-to-volume ratio exactly n, matching the Euclidean case. It includes invariance under origin-preserving affine maps and polar duality, an explicit integral formula in 2D that extends to non-symmetric sets, a recursive construction in higher dimensions via integration of lower-dimensional planar sections, and a perturbative result showing that generic changes to the Euclidean surface measure produce 4-fold symmetric sets at leading order in 2D. The 2D formula and the symmetry theorem are the clearest contributions; they are concrete enough to be checked directly and could be useful for anyone studying surface measures or small deformations of convex bodies. The invariance statements, if they hold without extra assumptions, would make the volume a reasonable candidate for affine-invariant problems. The recursive step is the soft spot. The abstract does not spell out why the n-dimensional volume, when restricted to any 2-plane, recovers the original 2D integral formula. If that fails, the whole object becomes dependent on the choice of embedding rather than intrinsic to the convex set, which would undermine its use as a canonical volume that preserves the ratio across dimensions. The preservation of P/V = n is built in by design, so the real work is in the invariances and the 2D analysis. This is for people in convex geometry and normed-space analysis who care about volumes that behave like Lebesgue measure under affine changes or duality. A reader interested in Alexandrov-type problems or explicit functionals on convex bodies would find the 2D parts worth reading. I would send it to peer review because the claims are specific and the potential gap in the recursion is something referees can test directly.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a definition of self-perimeter volume for the unit ball in n-dimensional normed spaces that is constructed to preserve the Euclidean relation P(B)/V(B)=n. It proves invariance of this volume under origin-preserving affine transformations and polar duality, derives an explicit integral formula for the self-perimeter in 2D (extended to non-centrally symmetric convex sets), extends the construction recursively to higher dimensions by integrating (n-1)-dimensional self-volumes of planar sections over the boundary, and poses an Alexandrov-type problem for the induced surface measure, with a perturbative analysis in 2D showing that generic perturbations of the Euclidean disk's surface measure produce 4-fold symmetric convex sets at leading order.

Significance. If the recursive definition is shown to be well-defined, consistent under subspace restrictions, and independent of embedding, the construction would supply a canonical volume in Minkowski geometry that retains the classical perimeter-volume scaling, with potential applications to isoperimetric inequalities and duality. The explicit 2D formula and the perturbative symmetry result constitute concrete, falsifiable contributions; the affine and polar invariances, if rigorously established without circularity, would strengthen the claim of naturality.

major comments (3)

[Recursive construction for n>2] Recursive construction (extension to R^n): the n-dimensional self-volume is defined by recursive boundary integration of (n-1)-volumes of planar intersections, starting from the explicit 2D formula. It is not shown that this recursion commutes with restriction to 2-planes, i.e., that the induced 2-volume on any subspace recovers the 2D self-perimeter formula. Without this consistency, the volume is embedding-dependent, undermining both the claim that it is intrinsic and the preservation of P/V=n across dimensions.
[Invariance proofs] Invariance under affine transformations and polar duality: these properties are asserted for the recursively defined volume, yet the proofs must explicitly verify that the recursive integration preserves the relation P/V=n after transformation; the construction appears tailored to enforce P/V=n by design, raising the question whether the invariances follow independently or by construction.
[Alexandrov-type problem and perturbative solutions] Perturbative result in 2D: the claim that generically any perturbation of the Euclidean disk's surface measure yields a 4-fold symmetric convex set at leading order requires a precise specification of the perturbation class (e.g., the Banach space of measures or support functions) and the precise meaning of 'generically'; the leading-order expansion should identify the symmetry-breaking modes that are excluded.

minor comments (2)

[Introduction and notation] Notation: the terms 'self-perimeter' and 'self-volume' should be clearly distinguished from classical perimeter and Lebesgue volume in the introduction and consistently used thereafter.
[Introduction] References: add citations to related constructions of volumes in normed spaces (e.g., dual mixed volumes, Petty projection bodies) to situate the self-perimeter volume within the existing literature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

Referee: Recursive construction for n>2: the n-dimensional self-volume is defined by recursive boundary integration of (n-1)-volumes of planar intersections... It is not shown that this recursion commutes with restriction to 2-planes, i.e., that the induced 2-volume on any subspace recovers the 2D self-perimeter formula. Without this consistency, the volume is embedding-dependent.

Authors: We acknowledge that the manuscript does not contain an explicit verification that the recursive definition is consistent with the 2D formula under restriction to arbitrary 2-planes. In the revised version we will add a lemma establishing this consistency: when the n-dimensional construction is restricted to any 2-plane, the resulting 2-volume coincides with the explicit integral formula derived for n=2. The proof proceeds by direct substitution of the recursive integral and reduction via Fubini-type arguments on the boundary measure. revision: yes
Referee: Invariance under affine transformations and polar duality: these properties are asserted for the recursively defined volume, yet the proofs must explicitly verify that the recursive integration preserves the relation P/V=n after transformation; the construction appears tailored to enforce P/V=n by design, raising the question whether the invariances follow independently or by construction.

Authors: The definition is chosen so that P/V=n holds by construction, but the affine and duality invariances are not automatic consequences of this scaling alone; they require that the recursive integration commutes with the respective transformations. The 2D case is proved directly from the integral formula. For the recursive step we will expand the argument in the revision to show explicitly that an origin-preserving affine map (or polar duality) applied to the n-dimensional body induces the corresponding map on each planar section, thereby preserving both the recursive structure and the P/V=n relation without circularity. revision: partial
Referee: Perturbative result in 2D: the claim that generically any perturbation of the Euclidean disk's surface measure yields a 4-fold symmetric convex set at leading order requires a precise specification of the perturbation class (e.g., the Banach space of measures or support functions) and the precise meaning of 'generically'; the leading-order expansion should identify the symmetry-breaking modes that are excluded.

Authors: We agree that the statement requires greater precision. In the revised manuscript we will specify the perturbation class as C^2-small perturbations of the support function (equivalently, measures with bounded total variation and C^1 density). 'Generically' will be clarified to mean an open-dense set in this topology. We will also include the explicit leading-order expansion in spherical harmonics, showing that all modes except the 4-fold symmetric ones (multiples of cos(4θ) and sin(4θ)) are annihilated by the first-order variational condition imposed by the self-perimeter functional. revision: yes

Circularity Check

1 steps flagged

Volume definition constructed to enforce P/V=n relation by design

specific steps

self definitional [Abstract]
"This paper introduces a natural definition for the volume of the unit ball in n-dimensional normed spaces R^n. This definition preserves the Euclidean relation P(B)/V(B)=n between the perimiter and the volume of the unit ball B in R^n."

The volume is introduced via a definition chosen to enforce the target ratio P/V=n. Preservation therefore holds by construction of the definition rather than emerging from independent axioms or prior results.

full rationale

The paper's central construction introduces a volume V for the unit ball specifically so that the Euclidean relation P(B)/V(B)=n is preserved. This makes the preservation a definitional feature rather than an independent derivation. The 2D explicit formula and recursive extension to higher dimensions are presented as natural, but the load-bearing step tying V to the perimeter ratio reduces to the choice of definition. Affine invariance and duality are shown after the definition is fixed, and the perturbative symmetry result appears to rest on separate analysis. No other self-citation or fitted-prediction circularity is exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based on abstract only; the construction relies on the recursive integration assumption and the choice of the natural definition without explicit free parameters or external evidence for invented concepts.

axioms (1)

domain assumption The recursive integration over the boundary using (n-1)-dimensional volumes of planar intersections defines a consistent volume measure preserving P/V=n
Invoked to extend the 2D formula to higher dimensions

invented entities (1)

self-perimeter volume no independent evidence
purpose: To assign volume to unit balls in normed spaces while preserving the Euclidean P/V relation
Newly introduced concept central to the paper

pith-pipeline@v0.9.0 · 5461 in / 1418 out tokens · 83174 ms · 2026-05-13T20:46:18.453920+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

C., & Thompson, A

´Alvarez Paiva, J. C., & Thompson, A. C. (1998).Volumes on normed and Finsler spaces. A Sampler of Riemann-Finsler Geometry, 50, 1-48

work page 1998
[2]

(1950).The geometry of Finsler spaces

Busemann, H. (1950).The geometry of Finsler spaces. Bulletin of the American Mathematical Society, 56(1), 5-16

work page 1950
[3]

D., & Thompson, A

Holmes, R. D., & Thompson, A. C. (1979).N-dimensional area and content in Minkowski spaces. Pacific Journal of Mathematics, 85(1), 77-110

work page 1979
[4]

(2014).Convex bodies: the Brunn-Minkowski theory(No

Schneider, R. (2014).Convex bodies: the Brunn-Minkowski theory(No. 151). Cambridge university press

work page 2014
[5]

Thompson, A. C. (1996).Minkowski geometry(Vol. 63). Cambridge University Press

work page 1996
[6]

Golab,Quelques probl` emes m´ etriques de la g´ eometrie de Minkowski, Trav

S. Golab,Quelques probl` emes m´ etriques de la g´ eometrie de Minkowski, Trav. de l’Acad. Mines Cracovie6(1932), 1–79

work page 1932
[7]

Gr¨ unbaum,Measures of symmetry for convex sets, Proc

B. Gr¨ unbaum,Measures of symmetry for convex sets, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1966, pp. 233–270

work page 1966
[8]

V. V. Makeev,On the self-circumference of a convex polygon, Journal of Mathematical Sciences116(2003), 3433–3435. 18

work page 2003
[9]

Ghandehari and H

M. Ghandehari and H. Martini,On self-circumferences in Minkowski planes, Extracta Mathematicae34(1) (2019), 73–89

work page 2019
[10]

A. I. Shcherba,Unit disk of smallest self-perimeter in the Minkowski plane, Journal of Mathematical Sciences140(2007), 589–592

work page 2007
[11]

Martini, K

H. Martini, K. J. Swanepoel, and G. Weiss,The geometry of Minkowski spaces - a survey. Part I, Expositiones Mathematicae19(2001), 97– 142. 19

work page 2001