Convexity and semiconvexity along vector fields

Bardi, Martino; Dragoni, Federica
November 2011
Calculus of Variations & Partial Differential Equations;Nov2011, Vol. 42 Issue 3/4, p405
Academic Journal
Given a family of vector fields we introduce a notion of convexity and of semiconvexity of a function along the trajectories of the fields and give infinitesimal characterizations in terms of inequalities in viscosity sense for the matrix of second derivatives with respect to the fields. We also prove that such functions are Lipschitz continuous with respect to the Carnot-Carathéodory distance associated to the family of fields and have a bounded gradient in the directions of the fields. This extends to Carnot-Carathéodory metric spaces several results for the Heisenberg group and Carnot groups obtained by a number of authors.


Related Articles

  • Local-Global Minimum Property in Unconstrained Minimization Problems. Burai, Pál // Journal of Optimization Theory & Applications;Jul2014, Vol. 162 Issue 1, p34 

    The main goal of this paper is to prove some new results and extend some earlier ones about functions, which possess the so-called local-global minimum property. In the last section, we show an application of these in the theory of calculus of variations.

  • Some Refinements of Ky Fan's and Sandor's Inequalities. Rooin, J. // Southeast Asian Bulletin of Mathematics;2004, Vol. 27 Issue 6, p1101 

    In this paper, using a refinement of unweighted Jensen's discrete inequality, we give some new refinements of Ky Fan's and Sandor's inequalities.

  • ON SOME INTEGRAL INEQUALITIES VIA h-CONVEXITY. TUNÇ, MEVLÜT // Miskolc Mathematical Notes;2013, Vol. 14 Issue 3, p1041 

    In this paper, we establish some new inequalities for class of SX (h, I) convex functions which are super-multiplicative or super-additive and nonnegative. And we also give some applications for special means.

  • EXTREME POINTS OF THE CLASS OF DISCRETE DECREASING FAILURE RATE LIFE DISTRIBUTIONS. Langberg, Naftali A.; Leon, Ramon V.; Lynch, James; Proschan, Frank // Mathematics of Operations Research;Feb80, Vol. 5 Issue 1, p35 

    Proposes that the class of discrete decreasing failure rate (DFR) life distributions is a convex class. Overview of the extreme points of the DFR class; Formula for the chain of inequalities for the proof of a lemma; Representation of discrete DFR life distributions as a mixture of extreme points.

  • Mathematical Properties of the Hyperbolicity of Circulant Networks. Hernández, Juan C.; Rodríguez, José M.; Sigarreta, José M. // Advances in Mathematical Physics;11/2/2015, p1 

    If X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangle   T={x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3], and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the...

  • Several new inequalities on operator means of non-negative maps and Khatri-Rao products of positive definite matrices. Zeyad Abdel Aziz Al-Zhour // Journal of King Abdulaziz University: Science;2014, Vol. 26 Issue 1, p21 

    In this paper, we provide some interested operator inequalities related with non-negative linear maps by means of concavity and convexity structure, and also establish some new attractive inequalities for the Khatri-Rao products of two or more positive definite matrices. These results lead to...

  • On the Structure of Cooperative and Competitive Solutions for a Generalized Assignment Game. Pablo Arribillaga, R.; Massó, Jordi; Neme, Alejandro // Journal of Applied Mathematics;2014, p1 

    We study cooperative and competitive solutions for a many-to-many generalization of Shapley and Shubik's (1971) assignment game. We consider the Core, three other notions of group stability, and two alternative definitions of competitive equilibrium. We show that (i) each group stable set is...

  • New Subclass of p-Valent Meromorphic Functions. Jingyu Yang; Shuhai Li // Pure Mathematics;Jul2012, Vol. 2 Issue 3, p117 

    In this paper, we introduce a subclass ?p,j(a, c, A, a m, ?) of the class ?p,j by use of Hadamard operator Lp(a, c) and subordination principle. The main objective of this paper is to provide coefficient inequality, inclusion properties, extreme points, convexity and starlike radius of this...

  • New inequalities of Hadamard type for quasi-convex functions. Özdemir, M. Emin; Yıldız, Çetin; Akdemir, Ahmet Ocak; Set, Erhan // AIP Conference Proceedings;8/10/2012, Vol. 1470 Issue 1, p99 

    In this paper some new Hadamard-type inequalities for functions whose second derivatives in absolute values are quasi-convex are established. Our results gives new estimations for quasi-convex functions.


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics