Stability of the Prékopa–Leindler Inequality

Definitions and mathematical statement

Let $n\geq 1$ be an integer, and let all integrals below be with respect to Lebesgue measure on ${\mathbb{R}}^n$. A centered probability density is a nonnegative measurable function $h$ satisfying \[ \int_{{\mathbb{R}}^n}h(x)\,{\,\mathrm{d}} x=1, \qquad \int_{{\mathbb{R}}^n}\|x\|h(x)\,{\,\mathrm{d}} x<\infty, \qquad \int_{{\mathbb{R}}^n}x h(x)\,{\,\mathrm{d}} x=0\in{\mathbb{R}}^n. \] Here the finite first moment makes explicit the usual Lebesgue-integral interpretation of the supplied phrase "zero mean"; the record does not state this moment condition separately. For probability densities $f,g$, define the truncated $L^1$ distance by \[ \delta_1(f,g)=\min\left\{1,\int_{{\mathbb{R}}^n}|f(x)-g(x)|\,{\,\mathrm{d}} x\right\}. \] The supplied conjectural question is whether, for every integer $n\geq1$, there exists a constant $\gamma_n>0$, depending only on $n$, such that the following holds for every $\lambda\in(0,1)$ and every triple of nonnegative measurable integrable functions $f,g,m$ on ${\mathbb{R}}^n$: if $f$ and $g$ are centered probability densities and \[ m\left(\lambda x+(1-\lambda)y\right)\geq f(x)^\lambda g(y)^{1-\lambda} \qquad\text{for every }x,y\in{\mathbb{R}}^n, \] then \[ \int_{{\mathbb{R}}^n}m(z)\,{\,\mathrm{d}} z\geq 1+\gamma_n\lambda\,\delta_1(f,g)^2. \] The pointwise hypothesis concerns the chosen representatives of the functions, not merely their almost-everywhere equivalence classes. The full parameter range $(0,1)$, the factor $\lambda$, and the absence of additional concavity or regularity hypotheses are retained exactly as supplied. Whether the source passage requires further restrictions has not been independently checked. This is a statement-only draft, not an assertion that the estimate holds or that its formulation has received mathematical review.

Short English statement

The supplied question asks whether the excess of the integral in the Prékopa--Leindler inequality uniformly controls the squared truncated $L^1$ distance between two centered probability densities, with coefficient $\gamma_n\lambda$. The constant must depend only on the dimension. The supplied partially solved status and source-verification label are curation claims, not independently verified conclusions.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001381$, supplied title "Stability of the Prékopa--Leindler Inequality," exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2009, and supplies no named proposer.
• Supplied source citation: Convex Geometry and its Applications (2010), Oberwolfach Reports, pp. 2851--2926, DOI 10.4171/owr/2009/53. https://doi.org/10.4171/owr/2009/53. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied literature-source label is "original Oberwolfach report 4136."
• Supplied literature reference, labeled "Ball--Boroczky stability result": https://arxiv.org/abs/0909.3738.
• Supplied literature reference, labeled "quantitative arbitrary-function stability": https://arxiv.org/abs/2201.11564.
• Supplied dated literature curation, checked at 2026-08-21: status "partially solved." Its assessment says that the report records only weaker stability exponents and special hypotheses, that later work proves quantitative stability in broader settings, and that the exact sharp conjectural estimate in the extracted passage is not established in full generality. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its verification note asserts that the official report states the domain, normalization, centering, pointwise Prékopa--Leindler hypothesis, truncated distance, and conjectured estimate reproduced in the record. That assertion is preserved as source metadata; no independent source inspection is claimed here.