Abstract: In this paper, some new concepts of geometrically relative convex sets and relative convex functions are defined. These new classes of geometrically relative convex functions unify several known and new classes of relative convex functions such as exponential convex functions. New Hermite-Hadamard type integral inequalities are derived for these new classes of geometrically relative c...

In convex programming, characterizations of the solution set in terms of the subdifferential have been investigated by Mangasarian. An invariance property of the subdifferential of the objective function is studied, and as a consequence, characterizations of the solution set by any solution point and any point in the relative interior of the solution set are given. In quasiconvex programming, h...

Mathematical models of phase transitions in solids lead to the variational problem, minimize R W (Du)dx where W has a multi-well structure: W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e., for K = denotes the (3 2)-matrix formed with the rst two columns of U i. We characterize generalized convex hulls, including the...

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the set polynomial degree $d\geq 2$ locally connected, topological dynamics can be completely described as quotient much simpler system: angle $d$-tupling on circle. transcendental entire function, less significant, but we may still ask for description system. To this end, introduce notion "...

For relatively hyperbolic groups, we investigate conditions under which the subgroup generated by two quasiconvex subgroups Q1 and Q2 is quasiconvex and isomorphic to Q1 ∗Q1∩Q2 Q2. Our results generalized known combination theorems for quasiconvex subgroups of word-hyperbolic groups. Some applications are presented. In addition, it is proved that the intersection of quasiconvex subgroups is qua...