Floating-Area and Centro-Affine Entropy Inequalities

Definitions and mathematical statement

The supplied record proposes three isoperimetric inequalities: \[ \Omega_1(K)\leq\Omega_1(C_K),\qquad E_{\mathrm{PW}}(K)\geq E_{\mathrm{PW}}(B_K),\qquad E^s(K)\geq E^s(C_K). \] Here the record calls $\Omega_1$ the floating area, $E_{\mathrm{PW}}$ the centro-affine entropy, and $E^s$ a spherical entropy analogue. The floating-area conjecture is explicitly proposed in every dimension $n\geq 2$, for every admissible $K$ in the intended geometric setting. The supplied text does not specify that setting or the admissibility conditions. It also does not specify the dimensions or admissible classes for the other two inequalities.

Unresolved definitions and hypotheses. No definitions or normalizations of $\Omega_1$, $E_{\mathrm{PW}}$, or $E^s$ are supplied. The comparison objects $C_K$ and $B_K$, their geometric types, and the quantities they must share with $K$ are undefined. In particular, the notation $B_K$ alone does not establish a choice of ball, normalization, or center. The occurrence of $C_K$ in both the floating-area and spherical assertions does not establish that the same construction is intended. Convexity, regularity, positivity, origin or hemisphere conditions, and conventions for infinite or undefined entropy values are not specified. Consequently, the displayed inequalities are the recorded conjectural targets; the supplied evidence does not determine complete definitions or a fully quantified mathematical conjecture.

Additional source qualifications. The original excerpt reports a boundary-integral representation of $E^s(K)$, involving a spherical curvature quantity, its logarithm, a boundary volume element, and a normalization involving $P^s(K^*)$. Its extracted formula is corrupted, so the integrand, measure, and normalization cannot be stated reliably. The excerpt also reports a logarithmic lower bound for $E^s(K)$ involving $P^s(K^*)$ and $P^s(K)$; the extracted layout does not reliably determine the formula. A further reported inequality is transcribed as \[ E^s(K^*)\geq E^s(C_{K^*}). \] Neither $K^*$, $P^s$, nor $C_{K^*}$ is defined in the supplied record, and the hypotheses of this reported result are absent. It is therefore retained as an attributed background assertion requiring verification. The excerpt's reference to $[5]$ is not accompanied by a bibliography identifying that numbered reference. The authors also express the belief that $E^s$ is a non-Euclidean analogue of both centro-affine entropy and Gaussian entropy; this is a stated interpretation, not a definition or theorem.

Short English statement

The problem asks whether floating area is bounded above by that of the intended comparison object, and whether centro-affine and spherical entropies are bounded below by their corresponding comparison values. The floating-area assertion explicitly concerns dimensions $n\geq 2$. Definitions of the functionals, comparison objects, and admissible bodies are missing, so these remain conjectural templates requiring source-based clarification and human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30006132$, source problem code catalogue entry, title Floating-Area and Centro-Affine Entropy Inequalities, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2024.
• Supplied report citation: Convex Geometry and its Applications (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/58, pp. 3301--3376. https://doi.org/10.4171/owr/2024/58. The citation's year 2025 and the report identifier containing 2024 are retained as supplied.
• Supplied literature source: Besau--Werner, 2024 preprint. https://arxiv.org/abs/2411.01631. The supplied record does not identify this link as the excerpt's numbered reference $[5]$.
• Supplied literature source: OWR 2024/58, open problems. https://ems.press/content/serial-article-files/50771.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved and states that the floating-area, centro-affine entropy, and spherical inequalities remain conjectural in their stated generality, with partial progress from dimension-two implications and related spherical $L_p$ inequalities. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. This is supplied verification metadata; no source inspection or mathematical verification is asserted by this draft.