We use asymptotic analysis to describe in amore systematic way the behavior at the infinity of functions in the convex and quasiconvex case. Starting from the formulae for the firstand second-order asymptotic function in the convex case, we introduce similar notions suitable for dealing with quasiconvex functions. Afterward, by using such notions, a class of quasiconvex vector mappings under wh...

In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p > 1, is the restriction to symmetric matrices of a 1-quasiconvex functio...

Dual characterizations of the containment of a convex set, defined by infinite quasiconvex constraints, in an evenly convex set, and in a reverse convex set, defined by infinite quasiconvex constraints, are provided. Notions of quasiconjugate for quasiconvex functions, λ-quasiconjugate and λsemiconjugate, play important roles to derive the characterizations of the set containments.

We prove that groups acting geometrically on δ-quasiconvex spaces contain no essential Baumslag-Solitar quotients as subgroups. This implies that they are translation discrete, meaning that the translation numbers of their nontorsion elements are bounded away from zero. The notion of translation numbers is used by many authors, such as J. Alonso and M. Bridson [1], G. Conner [2], S.M. Gersten a...

This class of functions strictly contains the class of convex functions defined on a convex set in a real linear space. See [8] and citations therein for an overview of this issue. Some recent studies have shown that quasiconvex functions have quite close resemblances to convex functions – see, for example, [4], [6], [7], [10] for quasiconvex and even more general extensions of convex functions...

Recently Hadamard-type inequalities for nonnegative, evenly quasiconvex functions which attain their minimum have been established. We show that these inequalities remain valid for the larger class containing all nonnegative quasiconvex functions, and show equality of the corresponding Hadamard constants in case of a symmetric domain.

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization techn...

We present necessary and sufficient optimality conditions for a problem with a convex set constraint and a quasiconvex objective function. We apply the obtained results to a mathematical programming problem involving quasiconvex functions.