Unspecified Log-Concave Measure Conjecture

Definitions and mathematical statement

Recoverable definition fragment. The supplied record describes a measure $\mu$ as being of the form $(\star)$ when it has density \[ e^{-w(|x|)},\qquad w:(0,\infty)\longrightarrow{\mathbb{R}}, \] where $w$ is increasing, followed by the unfinished condition "such that $t\mapsto w(e^t)$." The condition on this map is missing. Consequently, these data do not give a complete definition of $(\star)$. The fragment does not explicitly specify the ambient space or reference measure for this density, define $|x|$, distinguish strictly increasing from nondecreasing, or state normalization or integrability requirements. Since $w$ is defined only on $(0,\infty)$, the displayed density also leaves its value at the origin unspecified.

Mathematical statement unavailable. The excerpt begins with the isolated word "conjecture" and supplies no complete conjectural proposition. The properties $(B)$ and $(GZ)$, and the measure $\gamma$, are not defined. No assertion with recoverable hypotheses, conclusion, and quantifiers can therefore be stated from this record.

Supplied contextual claims. The excerpt historically reports few examples satisfying either $(B)$ or $(GZ)$ apart from $\gamma$. It lists Lebesgue measure restricted to a convex body $T$ as an example for $(GZ)$, and says no other examples for that property had previously been known. It attributes to Livshyts, reference [12], a version of $(GZ)$ for every even log-concave measure with a weakened exponent; the extracted exponents are corrupted and cannot be stated reliably. It attributes to Eskenazis, Nayar and Tkocz, reference [6], the property $(B)$ for certain Gaussian mixtures, including a displayed measure with parameter $0<p\leq1$; that measure's formula is also corrupted. The excerpt further attributes to references [14] and [13] the implication that the log-Brunn--Minkowski conjecture in dimension $n$ would imply both $(B)$ and $(GZ)$ for every even log-concave measure $\mu$ on ${\mathbb{R}}^n$, and cites [2] for the case $n=2$. The incomplete definition of $(\star)$ is associated with the unidentified work [4]. These are supplied contextual reports, not independently verified results or a substitute for the missing problem statement.

The supplied registry labels the record open, but its statement status is unrecoverable. Its supplied verification note says the official report places this fragment among proved results on log-concave measures and states no open problem. The open label thus does not establish the existence of an unresolved conjecture.

Short English statement

The record contains an incomplete definition of a class of measures and surrounding historical claims, but no recoverable conjecture. The missing condition on $t\mapsto w(e^t)$, undefined properties $(B)$ and $(GZ)$, and corrupted formulas prevent a precise problem statement. The supplied open label reflects uncertain curation; this statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004950$, exact source code catalogue entry, title "Unspecified Log-Concave Measure Conjecture." The supplied record belongs to the Oberwolfach Reports Open Problems collection; no proposer is identified, and the proposed year is 2021.
• Supplied source citation: Convex Geometry and its Applications (2022), Oberwolfach Reports, pp. 3187--3266, DOI 10.4171/owr/2021/59. Source URL and supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2021/59. Oberwolfach Reports document mathematical workshops and their open-problem discussions; this general provenance does not establish that the extracted fragment is an open problem.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. It labels the record open under an uncertainty rule because the extraction contains only the word "conjecture" and an incomplete measure-class definition. The supplied background requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note classifies the statement as unrecoverable and reports that the official report contains proved results at this location, without an open problem. It says no separate clean statement was promoted and the original record remains publicly visible for provenance. This note is supplied evidence, not a claim of source consultation here.
• Internal references [2], [4], [6], [12], [13], and [14] occur in the excerpt, but their bibliographic entries are not supplied. The excerpt names Livshyts in connection with [12] and Eskenazis, Nayar and Tkocz in connection with [6]. No authoritative citation for a complete conjectural statement is supplied.