The Gaussian chord Minkowski problem (Q6546781)

\textit{Y. Huang} et al. [Adv. Math. 385, Article ID 107769, 36 p. (2021; Zbl 1466.52010)] have first treated the Minkowski problem with Lebesgue measure replaced by Gaussian measure. \textit{E. Lutwak} et al. [Commun. Pure Appl. Math. 77, No. 7, 3277--3330 (2024; Zbl 07846680)] introduced chord measures and their corresponding Minkowski problems. The present paper combines both settings. For \(q>1\), define a functional on convex bodies \(K\subset{\mathbb R}^n\) (\(n\ge 2\)) by \N\[\NI_{\gamma,q}(K)= \int_K\int_K \frac{e^{-(|x|^2+|y|^2)/2}}{|x-y|^{n-q+1}}dxdy.\N\]\NIts differential, in the usual sense using Wulff shapes, is the \(q\)th Gaussian chord measure \(F_{\gamma,q}(K,\cdot)\), a finite Borel measure on the unit sphere \({\mathbb S}^{n-1}\). The following (partial) solution of the Gaussian chord Minkowski problem is provided. If \(\mu\) is a finite even Borel measure on \({\mathbb S}^{n-1}\) which is not concentrated on a closed hemisphere, then for any \(0<\tau< 1/(n+q-1)\) there exists a centrally symmetric convex body \(K\) such that \(\mu=cF_{\gamma,q}(K,\cdot)\), where \(c= I_{\gamma,q}(K)^{\tau-1}\). The constant \(c\) seems to be unavoidable, as so often when homogeneity is lacking. The proof uses the variational method, but requires elaborate estimates (involving an approximation argument), since the integrand of \(I_{\gamma,q}\) has a singularity if \(q<n+1\).