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Median porosity of sets is preserved under quasiconformal mappings.

2026-06-28 03:36 UTC pith:WUCVAC2D

load-bearing objection The paper shows median porosity (log-distance in BMO) is preserved by quasiconformal maps while weak porosity is not.

arxiv 2606.05034 v1 pith:WUCVAC2D submitted 2026-06-03 math.CA math.CVmath.MG

Median porosity is quasiconformally invariant

Tero Kilpel\"ainen , Antti V. V\"ah\"akangas This is my paper
classification math.CA math.CVmath.MG
keywords median porosityquasiconformal mappingsbounded mean oscillationquasiconformal invarianceporous setsweak porositydistance functionR^n
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

A set in Euclidean space is called median porous when the logarithm of its distance function has bounded mean oscillation. The paper proves that this property survives when the set is pushed forward by a quasiconformal mapping. Consequently median porosity qualifies as a quasiconformally invariant notion. By contrast the stronger condition of weak porosity is shown to fail the same invariance. Readers care because the result distinguishes a geometric feature that remains stable under the distortions commonly used to change coordinates in analysis.

Core claim

If a set E subset R^n satisfies the BMO condition on log dist(·,E), then for any quasiconformal map f the image f(E) satisfies the corresponding BMO condition on log dist(·,f(E)). The proof uses the standard analytic distortion estimates for quasiconformal mappings together with the equivalence of the BMO definition and the geometric definition of median porosity. The same argument does not hold for the stricter weak-porosity condition, which admits counterexamples under quasiconformal images.

What carries the argument

The BMO condition on the logarithm of the distance function to the set, which the paper treats as equivalent to the geometric notion of median porosity.

Load-bearing premise

The BMO definition of median porosity is equivalent to a purely geometric porosity condition and quasiconformal maps obey the usual integral distortion estimates.

What would settle it

Take a concrete median-porous set E in R^2, apply an explicit quasiconformal map such as a radial stretch, and verify whether the image satisfies the BMO condition on the log-distance function; failure on even one such pair would refute the invariance.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Quasiconformal images of median-porous sets remain median porous.
  • The class of median-porous sets is closed under quasiconformal equivalence.
  • Weak porosity is strictly stronger and fails to be invariant.
  • Median porosity can be used as a quasiconformally invariant descriptor of sets in geometric function theory.

Where Pith is reading between the lines

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

  • The invariance may allow median porosity to serve as a coordinate-independent test for porosity-type properties on quasiconformal images of domains.
  • It raises the question whether other BMO-based geometric conditions on distance functions are likewise invariant.
  • Counterexamples for weak porosity suggest that the precise oscillation threshold in the BMO definition is critical for the invariance to hold.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper defines median porosity of a set E ⊂ R^n via the condition that log(dist(·, E)) belongs to BMO(R^n). It proves that this property is preserved under quasiconformal mappings f : R^n → R^n, and therefore that median porosity is a quasiconformally invariant notion. By contrast, the stronger geometric condition of weak porosity is shown not to be preserved under the same class of mappings.

Significance. The result supplies a clean analytic characterization (via BMO) of a porosity notion that behaves well under the bi-Hölder distortion and change-of-variables properties of quasiconformal maps, while distinguishing it from a stronger notion that fails invariance. Such distinctions are useful in geometric measure theory and in the study of removable sets or dimension distortion under QC maps.

minor comments (3)
  1. §2, Definition 2.1: the precise normalization of the BMO seminorm (whether the mean is subtracted inside or outside the integral) should be stated explicitly, as it affects the constant in the subsequent distortion estimates.
  2. Theorem 3.2: the dependence of the BMO constant on the quasiconformal distortion K and the dimension n is not quantified; adding an explicit bound would strengthen the statement.
  3. The counter-example for weak porosity (presumably in §4) relies on a specific self-similar set; a brief remark on whether the construction generalizes to other dimensions would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proves quasiconformal invariance of median porosity (log-distance in BMO) by invoking standard analytic properties of QC maps (bi-Hölder distortion, BMO preservation under controlled change of variables) together with the known equivalence of the BMO definition to the geometric median-porosity notion. No equations reduce by construction to fitted inputs, no load-bearing self-citations appear, and the central claim is a direct preservation result under an external mapping class. The derivation is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background from analysis; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • standard math Standard properties of quasiconformal mappings in R^n
    Invoked implicitly to establish preservation of the BMO condition.
  • domain assumption Equivalence of BMO definition with median porosity
    Used to translate the geometric property into an analytic one.

pith-pipeline@v0.9.1-grok · 5581 in / 1066 out tokens · 50199 ms · 2026-06-28T03:36:15.766465+00:00 · methodology

0 comments
read the original abstract

A set in $\mathbb{R}^n$ is median porous if the logarithm of its distance function has bounded mean oscillation. We show that this property is preserved under quasiconformal mappings. In particular, median porosity is quasiconformally invariant. We also show that the stronger notion of weak porosity, by contrast, is not quasiconformally invariant.

discussion (0)

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Forward citations

Cited by 1 Pith paper

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

  1. One-sided median porous sets and one-sided Muckenhoupt distance functions

    math.CA 2026-07 unverdicted novelty 6.0

    One-sided median porosity of E is necessary and sufficient for d_E to the minus alpha to be in one-sided A_p for some alpha greater than zero when 1 less than p less than infinity, with new median characterizations of...

Reference graph

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