The Generalized Gaussian Minkowski Problem

This article delves into the \(L_p\) $L_p$ Minkowski problem within the framework of generalized Gaussian probability space. This type of probability space was initially introduced in information theory through the seminal works of Lutwak et al. (Ann Probab 32(1B):757–774, 2004, IEEE Trans Inf Theory 51(2):473–478, 2005), as well as by Lutwak et al. (IEEE Trans Inf Theory 58(3):1319–1327, 2012). The primary focus of this article lies in examining the existence of this problem. While the variational method is employed to explore the necessary and sufficient conditions for the existence of the normalized Minkowski problem when \(p \in \mathbb {R} \setminus \{0\}\) $p\in \mathbb{R}\backslash \{0\}$ , our main emphasis is on the existence of the generalized Gaussian Minkowski problem without the normalization requirement, particularly in the smooth category for \(p \ge 1\) $p\ge 1$ .