Cheeger-Space Criteria for Metric-Measure Spaces

Definitions and mathematical statement

Let $(X,d,m)$ denote the metric--measure space in the supplied record, where $d$ is a metric on $X$ and $m$ is a measure. The record does not specify the measurable structure or additional assumptions on $X$, $d$, or $m$.

Intended problem, with unresolved qualifications. Characterize when $(X,d,m)$ is a Cheeger space, and determine under which hypotheses the metric--measure construction discussed in the source produces the appropriate $p$-energy on $X$.

This is an incompletely specified research question. The supplied excerpt does not define "Cheeger space," specify the construction, give the range of $p$, identify the domain or codomain of the functions whose energy is considered, or define the criterion by which that energy is "appropriate." Consequently, it does not determine a precise universally quantified assertion or a specific energy identity. No completeness, separability, doubling, Poincaré inequality, measure regularity, or other analytic hypothesis is imposed here without source support.

The original excerpt also contains a corrupted formula involving $\operatorname{Lip} f(y)$, $|f(z)-f(y)|$, $d(y,z)<\delta$, and $\delta$. Its limiting operation and arrangement of suprema are not reliably specified, so it is not used to define a local slope or an energy. A precise statement requires the source's definitions and hypotheses for both the Cheeger-space condition and the proposed $p$-energy construction.

Short English statement

The record asks for criteria identifying Cheeger spaces and for conditions under which a construction using only their metric and measure gives an appropriate $p$-energy. The definitions, construction, exponent range, and comparison criterion are missing, so this draft preserves the intended question while leaving its precise formulation unresolved.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied permanent ID $30005984$, title "Cheeger-Space Criteria for Metric-Measure Spaces," source problem code catalogue entry. These are registry references supplied for this draft, not independently inspected files.
• Supplied report citation: Recent Developments in Dirichlet Form Theory and Related Fields (2025), Oberwolfach Reports, pp. 2425--2510, DOI 10.4171/owr/2024/42. Supplied source URL and literature-source link: https://doi.org/10.4171/owr/2024/42. The record gives proposed year 2024 and supplies no named proposer or identified speaker.
• Supplied literature reference: Cheeger, Differentiability of Lipschitz functions on metric measure spaces, https://doi.org/10.1007/s000390050094. The record does not identify a particular definition or result in this reference that fixes the intended formulation.
• The original excerpt attributes the construction to seminal work by Korevaar and Schoen, referenced there as [5], and to the construction of diffusion on $d$-sets by Kumagai and Sturm, referenced there as [9]. The supplied record does not include the corresponding bibliography entries.
• Supplied literature assessment, checked 2026-08-22: the record labels the problem open and describes Cheeger spaces as a defined and well-studied class, while observing that no particular metric--measure space is supplied and that membership and the relevant $p$-energy theory depend on hypotheses. This is supplied dated curation derived from the Oberwolfach record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.