Existence of Anisotropic Minkowski Content

Filip Fry\v{s}

arxiv: 2508.08156 · v5 · submitted 2025-08-11 · 🧮 math.CA

Existence of Anisotropic Minkowski Content

Filip Fry\v{s} This is my paper

Pith reviewed 2026-05-19 00:14 UTC · model grok-4.3

classification 🧮 math.CA

keywords Minkowski contentanisotropic perimetersets of finite perimetertopological boundaryouter Minkowski contentanisotropic Minkowski content

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The pith

For sets of finite perimeter the usual Minkowski content of the boundary equals the perimeter exactly when the anisotropic version equals half the sum of the anisotropic perimeters of the set and its complement.

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

The paper proves an if-and-only-if equivalence for sets E that already have finite perimeter: the Minkowski content of the topological boundary matches the perimeter of E if and only if the anisotropic Minkowski content of the same boundary equals half the sum of the anisotropic perimeter of E and the anisotropic perimeter of its complement. A direct consequence is that whenever the anisotropic outer Minkowski content exists for both E and its complement, the ordinary outer Minkowski content must also exist for both. This equivalence therefore supplies a bridge that lets existence results in the anisotropic setting imply existence in the standard Euclidean setting for the same class of sets.

Core claim

For a set E of finite perimeter the Minkowski content of its topological boundary coincides with the perimeter of E if and only if the anisotropic Minkowski content of the topological boundary coincides with one-half the sum of the anisotropic perimeter of E and the anisotropic perimeter of the complement of E. As an immediate corollary, the existence of the anisotropic outer Minkowski content for E together with the existence of the anisotropic outer Minkowski content for the complement guarantees the existence of the ordinary outer Minkowski content for E and for the complement.

What carries the argument

The anisotropic Minkowski content of the topological boundary, which replaces the Euclidean distance in the usual Minkowski content definition by an anisotropic norm and thereby measures boundary size in a direction-dependent way.

If this is right

Whenever the anisotropic Minkowski content exists and equals the averaged anisotropic perimeters, the ordinary Minkowski content exists and equals the perimeter.
Existence of anisotropic outer Minkowski content for both a set and its complement forces existence of ordinary outer Minkowski content for both.
Any criterion that guarantees the anisotropic outer content therefore automatically guarantees the ordinary outer content for the same sets.

Where Pith is reading between the lines

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

The same equivalence may let density or rectifiability results proved first in the anisotropic setting be transferred back to the standard setting.
The relation can be checked directly on model sets such as balls or half-spaces where both contents are computable by hand.
If the equivalence survives removal of the finite-perimeter hypothesis, it would extend the transfer of existence statements to a larger class of sets.

Load-bearing premise

The set E must already possess finite perimeter in the ordinary Euclidean sense.

What would settle it

A single explicit set of finite perimeter whose boundary has a well-defined Minkowski content equal to its perimeter but whose anisotropic Minkowski content fails to equal the average of the two anisotropic perimeters.

read the original abstract

This paper is devoted to the existence of anisotropic Minkowski content and anisotropic outer Minkowski content. Our result is that the Minkowski content of the topological boundary of a given set of finite perimeter $E$ coincides with the perimeter of $E$ if and only if the anisotropic Minkowski content of the topological boundary of $E$ coincides with half of the sum of the anisotropic perimeter of $E$ and the anisotropic perimeter of the complement of $E.$ As a consequence, we find that the existence of anisotropic outer Minkowski content of a given set of finite perimeter and its complement ensures the existence of outer Minkowski content of the set and its complement.

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

The paper gives a clean if-and-only-if equivalence between ordinary and anisotropic Minkowski contents of the boundary for finite-perimeter sets, plus a direct consequence for outer contents.

read the letter

The main thing to know is that this paper proves an equivalence: for a set E of finite perimeter, the Minkowski content of its topological boundary equals the perimeter of E if and only if the anisotropic Minkowski content equals half the sum of the anisotropic perimeter of E and the anisotropic perimeter of its complement. As a direct consequence, existence of the anisotropic outer Minkowski contents for both E and its complement implies existence of the ordinary outer Minkowski contents for both.

Referee Report

0 major / 2 minor

Summary. The paper proves that for a set E of finite perimeter, the (isotropic) Minkowski content of its topological boundary equals the perimeter of E if and only if the anisotropic Minkowski content of the boundary equals half the sum of the anisotropic perimeters of E and its complement. As a consequence, existence of the anisotropic outer Minkowski contents for both E and E^c implies existence of the corresponding isotropic outer Minkowski contents.

Significance. If correct, the equivalence provides a precise bridge between isotropic and anisotropic Minkowski contents restricted to the class of finite-perimeter sets, together with a direct existence-transfer result. The restriction to sets already possessing finite perimeter is a strength, as it avoids overgeneralization. The manuscript ships a clean logical implication from the iff statement to the existence consequence.

minor comments (2)

[Abstract] The abstract and introduction should explicitly state the ambient dimension n and the precise definition of the anisotropic norm φ used throughout (e.g., whether it is assumed convex, 1-homogeneous, etc.).
[Section 2] Notation for the anisotropic perimeter Per_φ(E) and the anisotropic Minkowski content should be introduced with a short reminder of their definitions in §2 to aid readers unfamiliar with the anisotropic setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. We appreciate the recognition of the equivalence result and its consequence for existence of Minkowski contents in the finite-perimeter setting.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from finite-perimeter properties

full rationale

The central result is an explicit if-and-only-if equivalence relating the standard Minkowski content of the topological boundary to the perimeter of E, and the anisotropic version to half the sum of anisotropic perimeters of E and its complement, for sets E already known to have finite perimeter. The existence consequence for outer contents follows directly as a logical implication of this equivalence when the anisotropic quantities exist. No equations reduce a claimed prediction to a fitted input by construction, no self-citation is invoked as a load-bearing uniqueness theorem, and no ansatz is smuggled via prior work. The statement is carefully scoped to the BV/finite-perimeter class, so the derivation chain rests on standard properties of sets of finite perimeter rather than on redefinition or renaming of the target quantities themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard theory of sets of finite perimeter and anisotropic perimeters but introduces no new free parameters, axioms beyond domain standards, or invented entities.

axioms (1)

domain assumption E is a set of finite perimeter in Euclidean space
The entire statement is conditioned on this property of E.

pith-pipeline@v0.9.0 · 5615 in / 1220 out tokens · 32664 ms · 2026-05-19T00:14:05.001646+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Minkowski content of the topological boundary of a given set of finite perimeter E coincides with the perimeter of E if and only if the anisotropic Minkowski content ... coincides with half of the sum of the anisotropic perimeter of E and ... complement
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Proof relies on the Besicovitch covering theorem and [LV, Lemma 3.3]

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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