In this paper we investigate the optimal problem for the L(p )mixed chord integral. The existence of the L-p chord integral-Petty bodies is established, and the L-p geominimal chord integral is proposed and its properties, such as invariance under orthogonal matrices, homogeneity, isoperimetric-type inequalities and cyclic-type inequalities, which are provided as well.

In this paper, the L_q-Minkowski problem of anisotropic p-torsional rigidity is considered. The existence of the solution of the L_q-Minkowski problem of anisotropic p-torsional rigidity with 0<q<1 and 1<q≠p/p-1+n is given.

In this paper, we confirm the existence of asymmetric smooth solutions to the L (p)-Gauss Minkowski problem for p > 0 by a class of inverse Gauss curvature flows. Furthermore, asymmetric weak solution for p > 0 is obtained by a parabolic approximation method. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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.

In this paper, we introduce the L-p q-torsional measure for p is an element of R and q > 1 by the L-p variational formula for the q-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the L-p Minkowski problem with respect to the q-torsional rigidity for discrete measures and general measures when 0 < p < 1 and q > 1.

As we all know, the Minkowski type problem is the cornerstone of the Brunn-Minkowski theory in Euclidean space. The Heisenberg group as a sub-Riemannian space is the simplest non-Abelian degenerate Riemannian space that is completely different from a Euclidean space. By analogy with the Minkowski type problem in Euclidean space, the Minkowski type problem in Heisenberg groups is still open. In the present paper, we develop for the first time a sub-Riemannian version of Minkowski type problem in the horizontal distributions of Heisenberg groups, and further give a positive answer to this sub-Riemannian Minkowski type problem via the variational method.

The existence of symmetric weak solutions for p ≥ 1 and p ≤ 0 and asymmetric weak solutions for p ≥ 1 to the Lp-Gauss Minkowski problem, introduced in [Calc.Var., 61:28 (2022)], is proved by variational method and degree-theoretic method in [29] and [11], respectively. However, the smoothness of the solution is still a blank. In this paper, we confirm the existence of asymmetric smooth solutions to the Lp-Gauss Minkowski problem for p > 0 by a class of inverse Gauss curvature flows. Furthermore, asymmetric weak solution for p> 0 is obtained by a parabolic approximation method.

The present paper introduces a new class of geometric measures, the k-th (p,q)-mixed curvature measures, and a natural correspondence-(p,q)-Christoffel-Minkowski problem is proposed. The (p,q)-Christoffel-Minkowski problem posed here can be regarded as a natural generalization of the L_p Christoffel-Minkowski problem and Lp dual Minkowski problem. We investigate and arrive at the existence of smooth solution to the (p,q)-Christoffel-Minkowski problem by a type of expanding curvature flow. Furthermore, the uniqueness result of solutions to the (p,q)-Christoffel-Minkowski problem shall be discussed.

In this paper, we investigate an expanding flow of a convex hypersurface in Euclidean space Rn, with speed depending on the k-th symmetric polynomial of the principal curvature radii, support function and radial function. We prove that the flow has smooth and uniformly convex solution for all time and converges smoothly to a homothetic self-similar solution satisfies a nonlinear PDE of Hessian type. As an application of this expanding flow, we further confirm the existence of non-even smooth solution to the Christoffel-Minkowski type problem with respect to M(p,q),k(K,{\omega}), which can be regarded as a natural generalization of the celebrated works by Huang and Zhao, Guan and Xia, Chen and Ma in [4, 11, 15, 16].

Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C 0 -estimate technique under weaker conditions. As an application of this inverse Gauss curvature flow, the present paper first arrives at a non-even smooth solution to the Orlicz Minkowski problem.

The conformai properties of complex Finsler metrics are studied. We first give a characterization of a compact complex Finsler manifold to be globally conformai Kahler. By considering the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes, we then study the variational properties of Kahler Finsler metrics. By studying the spectral properties of two average metrics, the stabilities of critical Kähler Finsler metrics are verified. Finally, a Yamabe type problem for mean holomorphic Ricci curvature is considered, and a partial existence result is obtained.

We obtain a partial parallelism of the complex structure on Kähler Finsler manifolds. As applications, we prove Synge-Tsukamoto theorem and Bonnet-Myers theorem for positively curved Kähler Finsler manifolds. Moreover, we generalize a comparison theorem due to Ni-Zheng by introducing the notion of orthogonal Ricci curvature to Kähler Finsler geometry.

The classical Yamabe problem in Riemannian geometry states that every conformal class contains a metric with constant scalar curvature. In Finsler geometry, the C-convexity is needed in general. In this paper, we study the strong C-convexity of Randers metrics, and provide a result on the Yamabe problem for the metrics of Randers type.

On a Riemannian manifold, a Liouville transformation is a conformal change which preserves the Ricci tensor. In Finsler geometry, there are various types of Ricci curvature. By adding a Landsberg curvature term to the classical Ricci curvature, we consider the conformal transformations on a Finsler manifold such that this modified Ricci curvature is preserved. We prove that such conformal transformations are homothetic if the space is C-convex. The conformal rigidity for Landsberg surfaces is also obtained.

In this paper we first obtain the existence of smooth solutions to Orlicz-Aleksandrov problem via a Gauss-like curvature flow.

In this paper, we introduce the concept of Lp-mixed radial Blaschke–Minkowski homomorphisms for star bodies. Further, associated with dual quermassintegrals, we give an affirmative answer and two negative answers of the Busemann–Petty problem for the Lp-mixed radial Blaschke–Minkowski homomorphisms.

In 2006, Schuster introduced the concept of Blaschke-Minkowski homo-morphism of convex bodies. In this paper, we introduce the L-p-mixed Blaschke-Minkowski homomorphism in L-p-Brunn-Minkowski theory. We then further study the Shephard type problem involving an affirmative answer and two negative answers for L-p-mixed Blaschke-Minkowski homomorphism.

Lutwak, Yang and Zhang [23] introduced the concept of Lp dual curvature measure for convex bodies and star bodies, and studied the Minkowski problem. We in this paper establish a new unified concept, in briefly, the (p,q)-mixed quermassintegrals, via (p,q)-dual mixed curvature measure, and further have a deep discussion on Minkowski problem with respect to the (p,q)-dual mixed curvature measure. By the way, we derive at some important properties and geometric inequalities for (p,q)-mixed quermassintegrals.

In this paper, we study conformally flat (alpha, beta)-metrics in the form F = alpha(1 Sigma(m )(j=1)a(j) (beta/alpha)(j)) with m >= 2, where a is a Riemannian metric and beta is a 1-form on a smooth manifold M. We prove that if such conformally flat (alpha, beta)-metric F is of weakly isotropic scalar curvature, then it must has zero scalar curvature. Moreover, if a(m-1)a(m) not equal 0, then such metric is either locally Minkowskian or Riemannian.