Path of quasi-means as a geodesic

As a generalization of the Hiai-Petz geometries, we discuss two types of them where the geodesics are the quasi-arithmetic means and the quasi-geometric means respectively. Each derivative of such a geodesic might determine a new relative operator entropy. Also in these cases, the Finsler metric can be induced by each unitarily invariant norm. If the norm is strictly convex, then the geodesic is the shortest. We also give examples of the shortest paths which are not the geodesics when the Finsler metrics are induced by the Ky Fan k-norms.     

Stable Norms of Surfaces: Local Structure of the Unit Ball at Rational Directions

Let (M, g) be a closed orientable surface, equipped with a smooth Finsler metric. The metric induces a norm on the real homology of M, called a stable norm. We show this norm is neither strictly convex, nor smooth. Submitted: December 1996, final version: June 1997     

On the maximum principle on complete Finsler manifolds

In this paper, we generalize Omori–Yau maximum principle to Finsler geometry. As an application, we obtain some Liouville-type theorems of subharmonic functions on forward complete Finsler manifolds.     

Local gradient estimate for harmonic functions on Finsler manifolds

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorems on noncompact Finsler manifolds with nonnegative Ricci curvature.     

On the convexifiability of pseudoconvex C2-functions

We present new criteria that characterize functions which are convex transformable by a suitable strictly increasing function. We concentrate on twice continuously differentiable pseudoconvex and strictly pseudoconvex functions, and derive conditions which are both necessary and sufficient for these functions to be convex transformable.     

Hypersurfaces of Constant Curvature in Hyperbolic Space I

We investigate the problem of finding, in hyperbolic space, a complete strictly convex hypersurface which has a prescribed asymptotic boundary at infinity and which has some fixed curvature function being constant. Our results apply to a very general class of curvature functions.     

New formulas for the Legendre-Fenchel transform

We present new formulas for the Legendre-Fenchel transform of functions. They concern the following three operations: inverting a strictly monotone convex function, post-composing an arbitrary function with a strictly monotone concave function, multiplying two positively valued strictly monotone convex functions.     

Convex underestimation for posynomial functions of positive variables

The approximation of the convex envelope of nonconvex functions is an essential part in deterministic global optimization techniques (Floudas in Deterministic Global Optimization: Theory, Methods and Application, 2000). Current convex underestimation algorithms for multilinear terms, based on arithmetic intervals or recursive arithmetic intervals (Hamed in Calculation of bounds on variables and underestimating convex functions for nonconvex functions, 1991; Maranas and Floudas in J Global Optim 7:143–182, (1995); Ryoo and Sahinidis in J Global Optim 19:403–424, (2001)), introduce a large number of linear cuts. Meyer and Floudas (Trilinear monomials with positive or negative domains: Facets of convex and concave envelopes, pp. 327–352, (2003); J Global Optim 29:125–155, (2004)), introduced the complete set of explicit facets for the convex and concave envelopes of trilinear monomials with general bounds. This study proposes a novel method to underestimate posynomial functions of strictly positive variables.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.     

Convex functions, subdifferentiability and renormings

This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces. Supported by NSFC     

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric.This result together with Zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.     

Convexity and Some Geometric Properties

The main goal of this paper is to present results of existence and nonexistence of convex functions on Riemannian manifolds, and in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Riemannian manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete manifolds, and we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result, we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.     

强拟\alpha-预不变凸函数

研究了一类重要的广凸函数------强拟$\alpha$-预不变凸函数,讨论了它与拟\,$\alpha$-预不变凸函数、严格拟\,$\alpha$-预不变凸函数及半严格拟\,$\alpha$-预不变凸函数之间的关系,并在中间点的强拟\,$\alpha$-预不变凸性下得到了它的三个重要的性质定理,同时给出了强拟\,$\alpha$-预不变凸函 数在数学规划中的两个重要应用,这些结果在一定程度上完善了对强拟\,$\alpha$-预不变凸函数的研究.     

Warped products of Hadamard spaces

In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved by warped products with convex warping functions. Received: 9 January 1998/ Revised version: 12 March 1998     

Minkowskian geometry,convexity conditions and the parallel axiom

The problem of characterising Minkowskian spaces is an important problem of that branch of differential geometry in which spaces more general than the complete Riemann and Finsler spaces are studied axiomatically using synthetic geometric methods. The fundamental theorem in this field is the result that a Desarguesian straightG-space in which the parallel axiom holds and the spheres are convex is Minkowskian. However the question as to whether the hypothesis of the space being Desarguesian is necessary or not has remained unsolved for over forty years. It is therefore natural to investigate conditions stronger than the mere convexity of spheres. In this paper such geometric conditions derived from functions which measure the distance between lines and points on lines are studied. Besides characterising the Minkowskian spaces these investigations also bring out the interplay between the parallel axiom and the convexity and linearity conditions.     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

对称弧式连通凸多目标半无限规划的最优性

以弧式连通函数和对称梯度为基础,研究新函数在多目标半无限规划下的最优性理论.定义了一类新的弧式连通函数,对称弧式连通函数、对称拟弧式连通函数、对称弱拟弧式连通函数、对称伪弧式连通函数、对称严格伪弧式连通函数,讨论了这些函数在多目标半无限规划下的最优性.给出更加广义的弧式连通函数,将它们运用到多目标半无限规划.     

Design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities

In this paper, design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities is presented. A new synchronization criterion is proposed based on Lyapunov functions with quadratic form of states and nonlinear functions of the systems. Sector and slope bounds are employed to the Lyanunov–Krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained. The criteria is given in terms of linear matrix inequalities (LMIs) by using Finsler’s lemma. A numerical example is provided to illustrate the effectiveness of the method.     

The Convexity and the Gaussian Curvature Estimates for the Level Sets of Harmonic Functions on Convex Rings in Space Forms

In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary.     

Filling minimality of finslerian 2-discs

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes-Thompson volume. This implies a generalization of Pu’s isosystolic inequality to Finsler metrics, both for the Holmes-Thompson and Busemann definitions of the Finsler area.