Log-Concave Busemann-Petty Problems

Definitions and mathematical statement

Let $K,L\subset{\mathbb{R}}^n$ be origin-symmetric convex bodies, meaning compact convex sets with nonempty interior satisfying $K=-K$ and $L=-L$. Write \[ S^{n-1}=\{\theta\in{\mathbb{R}}^n:|\theta|=1\},\qquad \theta^\perp=\{x\in{\mathbb{R}}^n:\langle x,\theta\rangle=0\},\qquad tK=\{tx:x\in K\}. \] For a measure with specified density $d\mu(x)=f(x)\,dx$, define its induced hyperplane measure by \[ \mu_H(A)=\int_A f(x)\,d\operatorname{vol}_{n-1,H}(x),\qquad A\subset H, \] where $\operatorname{vol}_{n-1,H}$ is Euclidean Lebesgue measure on the hyperplane $H$. The section notation in the supplied statement is interpreted here as this induced measure. This interpretation requires confirmation: ordinary ambient $\mu(A)$ is zero for every hyperplane subset $A$ when $\mu$ has a Lebesgue density.

A nonnegative density $f$ is log-concave if \[ f((1-\lambda)x+\lambda y)\geq f(x)^{1-\lambda}f(y)^\lambda \quad(x,y\in{\mathbb{R}}^n,\ 0<\lambda<1). \] Rotation invariance means that the specified density has the form $f(x)=g(|x|)$. The phrase "strictly decreasing density" is understood to refer to the radial profile $g$; the supplied record does not specify whether strict decrease is required on all radii or only on its positive support.

Question (1). For each $n\in\{5,6\}$, let $\mu$ be a rotation-invariant log-concave measure with strictly decreasing radial density, and let $K,L\subset{\mathbb{R}}^n$ be origin-symmetric convex bodies. Is the implication \[ \left[ \forall t>0\ \forall\theta\in S^{n-1},\quad \mu_{\theta^\perp}(tK\cap\theta^\perp) \leq \mu_{\theta^\perp}(tL\cap\theta^\perp) \right] \quad\Longrightarrow\quad \mu(K)\leq\mu(L) \] valid?

For the second question, the Minkowski functional of an origin-symmetric star body $M$, with continuous positive radial function and the origin in its interior, is \[ \|x\|_M=\inf\{a>0:x\in aM\}. \] This functional need not be a norm unless $M$ is convex. In dimensions $n\geq2$, a classical intersection body $M=IB$ of an origin-symmetric star body $B$ satisfies \[ \rho_M(\theta)=\operatorname{vol}_{n-1,\theta^\perp}(B\cap\theta^\perp), \qquad \rho_M(\theta)=\sup\{r\geq0:r\theta\in M\}. \] The term "intersection body" can also denote the broader class obtained by radial limits of such bodies. The supplied record does not specify which convention is intended.

Question (2), with unresolved scope. In arbitrary dimension, let $\mu$ be log-concave with specified density \[ d\mu(x)=\exp\left(-\phi(\|x\|_M)\right)\,dx, \] where $M$ is an intersection body. For every pair of origin-symmetric convex bodies $K,L$, does the same universally quantified section comparison displayed in Question (1) imply $\mu(K)\leq\mu(L)$? Here the section comparison and the conditions on $K,L$ are inherited from the supplied reconstructed statement. The supplied original excerpt does not state these hypotheses independently. It also does not specify the domain, allowed values, monotonicity, or strict monotonicity of $\phi$, nor whether strict radial decrease from Question (1) is retained. Log-concavity of $\mu$ is explicitly required by that excerpt. Euclidean rotation invariance is not imposed in this displayed formulation, since the density is expressed using $M$; whether any additional invariance was intended remains unresolved. The admissible dimension range and the intended intersection-body convention also require confirmation. These qualifications prevent a unique fully specified formulation of Question (2).

Short English statement

The first question asks whether comparison of all central hyperplane sections at every common positive dilation forces comparison of the full measures in dimensions $5$ and $6$. The second asks for an analogous implication for log-concave densities expressed through an intersection body's Minkowski functional in arbitrary dimension. The section-measure convention and several hypotheses of the second question require confirmation. Both questions are supplied as open; this draft does not independently verify their status.

Sources

• Local UnsolvedMath registry record, permanent ID $30006130$, source problem code catalogue entry, title Log-Concave Busemann-Petty Problems; supplied from the repository registry associated with data/unsolvedmath/problems.json and catalog/problems.json. No individual proposer is supplied. The record gives proposed year $2024$.
• 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 $2025$ designation and its $2024$ DOI/report designation are retained as supplied.
• 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 questions open, reports Gaussian counterexamples for $n\geq7$, and describes the requested $n=5,6$ and general intersection-body cases as unresolved. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and states that it was reconstructed and checked against the official report. Those are supplied provenance claims; no source visit or independent verification is asserted here. The supplied original excerpt contains two questions whose section-comparison context is omitted from that excerpt.