Abstract A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $\varphi $ and the perturbation of $\varphi $ via the infimal convolution. This formula naturally leads to a Borel measure on $\mathbb{R}^{n}$ and a Borel measure on the unit sphere $S^{n-1}$. The resulting Borel measure on $\mathbb{R}^{n}$ will be called the Euclidean Gaussian moment measure of the convex function $\varphi $, and the related Minkowski-type problem will be studied. A solution to the newly posed Minkowski problem will be solved under some mild and natural conditions on the pre-given measure.

The $$L_q$$ Minkowski problem for p-harmonic measure

On the number of solutions to the planar dual Minkowski problem

Abstract To every log-concave function f one may associate a pair of measures $$(\mu _{f},\nu _{f})$$ ( μ f , ν f ) which are the surface area measures of f . These are a functional extension of the classical surface area measure of a convex body, and measure how the integral $$\int f$$ ∫ f changes under perturbations. The functional Minkowski problem then asks which pairs of measures can be obtained as the surface area measures of a log-concave function. In this work, we fully solve this problem. Furthermore, we prove that the surface area measures are continuous with respect to a suitable topology: If $$f_{k}\rightarrow f$$ f k → f , then $$\left( \mu _{f_{k}},\nu _{f_{k}}\right) \rightarrow \left( \mu _{f},\nu _{f}\right) $$ μ f k , ν f k → μ f , ν f in a corresponding sense. Finding the appropriate mode of convergence of the pairs $$\left( \mu _{f_{k}},\nu _{f_{k}}\right) $$ μ f k , ν f k sheds a new light on the construction of functional surface area measures. To prove this continuity theorem we associate to every convex function a new type of radial function, which seems to be an interesting construction on its own right. Finally, we prove that the solution to the functional Minkowski problem is continuous in the data, in the sense that if $$\left( \mu _{f_{k}},\nu _{f_{k}}\right) \rightarrow \left( \mu _{f},\nu _{f}\right) $$ μ f k , ν f k → μ f , ν f then $$f_{k}\rightarrow f$$ f k → f up to translations.

Minkowski problems of centro-section measures

Flow by Gauss curvature to the Orlicz Minkowski problem for q-torsional rigidity

A class of generalized chord Minkowski problems

The Orlicz chord Minkowski problem for polytopes

Positive periodic solutions to the planar L dual Minkowski problem in the critical case

Periodic solutions to the planar $$L_p$$ dual Minkowski problem with sign-changing data

The L_{p} -Minkowski problem deals with the existence of closed convex hypersurfaces in \mathbb{R}^{n+1} with prescribed p -area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. Existence of solutions has been obtained in the sub-critical case {p>-n-1} , but the problem remains open in the super-critical case {p<-n-1} . In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids. Our results show that the L_{p} -Minkowski problem admits a solution in both the sub-critical and super-critical cases but does not have a solution in general in the critical case.

The Orlicz Minkowski problem for logarithmic capacity seeks to determine the necessary and sufficient conditions for a given finite Borel measure, such that it is the Orlicz logarithmic capacitary measure of a convex body. The Orlicz Minkowski problem for logarithmic capacity includes the Minkowski problem for logarithmic capacity and the [see formula in PDF] Minkowski problem for logarithmic capacity as special cases. The discrete case has been solved by the researchers. In this paper, we solve the Orlicz Minkowski problem for logarithmic capacity with respect to general Borel measures by applying an approximation scheme.

In a previous paper [H. Li and B. Xu, arXiv:2211.06875 (2022)], the first author and Xu introduced and studied the horospherical p-Minkowski problem in hyperbolic space Hn+1. In particular, they established the uniqueness result for solutions to this problem when the prescribed function is constant and p ≥ −n. This paper focuses on the isotropic horospherical p-Minkowski problem in hyperbolic plane H2, which corresponds to the equation φ−pφθθ−φθ2/(2φ)+(φ−φ−1)/2=γ on S1, where γ is a positive constant. We provide a classification of solutions for p ≥ −7, as well as a nonuniqueness result of solutions for p < −7. Furthermore, we extend this problem to the isotropic horospherical q-weighted p-Minkowski problem in hyperbolic plane and derive some uniqueness and nonuniqueness results.

The Orlicz Gaussian Minkowski Problem for General Measures

Abstract We derive the stability result of the dual curvature measure with near-constant density in the even case. As an application, the existence and uniqueness of solutions to the even dual Minkowski problem for positive indices in $\mathbb R^{n+1}$ are obtained with $n\geq 1$, provided the density of the given measure is close to 1 in the $C^{\alpha }$ norm with $\alpha \in (0,1)$.

The discrete L Minkowski problem for log-capacity for p < 0

In this paper, we prove the existence of convex solutions to a Monge–Ampère equation of entire type derived by a weighted version of the classical Minkowski problem.

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Ampère equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and sufficient condition for existence and uniqueness) in dimension 2 is presented. In higher dimensions, partial results are demonstrated.

Abstract In this paper, we investigate two anisotropic Gaussian curvature flows. Through establishing the long-time existence and congergence for these two flows, we derive the existence results for the Orlicz-Gaussian Minkowski problem in both origin-symmetric and general convex body settings.

This paper explores the p -capacitary Orlicz–Minkowski problem. Note that the p -capacitary Orlicz–Minkowski problem can be converted equivalently to a Monge–Ampère type equation in the smooth case: \tag{$\star$} f\phi(h_{K}) |\nabla\Psi|^{p}=\tau G for p\in (1,n) and some constant \tau>0 , where f is a positive function defined on the unit sphere \mathcal{S}^{n-1} , \phi is a continuous positive function defined in (0,+\infty) , and G is the Gauss curvature.We confirm for the first time the existence of smooth solutions to the p -capacitary Orlicz–Minkowski problem for p\in (1,n) using a class of inverse Gauss curvature flows which converges smoothly to the solution of equation (\star) . Moreover, we prove uniqueness for equation (\star) in a special case.