Detecting Geometric Scales in Fractal Measurements

Definitions and mathematical statement

Let $X$ denote a space whose microscopic and macroscopic geometries differ, as described qualitatively in the supplied source record.

Scale-detection problem. Develop an intrinsic procedure that detects the appropriate scale at which to measure volume, perimeter, and isoperimetric behavior on $X$.

Unresolved formulation. The supplied record does not specify a class of admissible spaces, the structures given on $X$, definitions of volume or perimeter, or a mathematical criterion for distinguishing microscopic from macroscopic geometry. It also does not define an intrinsic procedure, its required output, or what makes a measurement scale correct. In particular, it is unspecified whether a single scale should serve all three measurements, whether scales may depend on location or on the set being measured, and which isoperimetric inequality or scale-dependent behavior is sought. Consequently, this is a qualitative research question; the supplied evidence does not determine a fully quantified mathematical assertion.

The extracted original text mentions fractal spaces and includes a figure caption describing an approximation of a Sierpinski gasket cable system. This supplies context, but does not establish that such cable systems are the intended exclusive domain or provide additional hypotheses.

Short English statement

Find an intrinsic way to identify suitable measurement scales in spaces with different small-scale and large-scale geometries, particularly in the fractal setting mentioned in the source. The admissible spaces, measurement definitions, and criterion for a correct scale remain unspecified, so a precise theorem statement requires further source clarification.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30006505$, title Detecting Geometric Scales in Fractal Measurements, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The registry records proposed year $2025$; its proposer field is empty.
• Supplied report citation: Functional Inequalities: Geometric Calculus meets Stochastic Analysis (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/54, pp. 2917--2994. https://doi.org/10.4171/owr/2025/54. The supplied original extraction associates the question with Patricia Alonso Ruiz, joint work with Fabrice Baudoin, under the title An isoperimetric inequality for (fractal) spaces with different micro- and macrostructure. This attribution does not establish a separately identified problem proposer.
• Supplied literature source: Alonso Ruiz--Baudoin, Besov classes on Dirichlet spaces. https://arxiv.org/abs/1903.10078.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that existing fractal-space isoperimetric results require specified structures and scales, and that no general theorem answering the extracted scale-detection question was found. The supplied checked-source list consists of Oberwolfach Report 2025/54 and the Alonso Ruiz--Baudoin reference above. This is supplied dated literature triage, not independently verified here; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that only clear formula delimiters, notation, grammar, and typography were repaired while mathematical content and scope were preserved. These are supplied source-processing claims, not a mathematical verification of this draft.