On the spherical quasi-convexity of quadratic functions

Abstract

In this paper the spherical quasi-convexity of quadratic functions on spherically self-dual sets is studied. Sufficient conditions for spherical quasi-convexity on self-dual sets are presented. A partial characterization of spherical quasi-convexity on spherical Lorentz sets is given and some examples are provided.

On the Spherical Quasi-convexity of Quadratic Functions on Spherically Subdual Convex Sets

“…In this section, we present partial conditions characterizing the spherical quasiconvexity of quadratic functions on spherically subdual convex sets associated to subdual cones. The results obtained generalize the corresponding ones obtained in [10,Sect. 4.1].…”

“…In particular, several examples are presented. This paper continues the study of [10], which can be regarded as premier study about the topic of quasi-convexity of quadratic functions on the Euclidean space. The main literature about the quasiconvexity of quadratic functions on Euclidean convex sets includes, but it is not limited to [11][12][13][14][15].…”

“…In this section, we recall the concept of spherically quasi-convex functions on spherically convex sets and we present a basic characterization of them; for more details, see [10]. For this concept and its properties in Euclidean space, see [23].…”

“…For the simplicity of notations, we will denote the extended values by ϕ A (x) too, but the quasiconvexity of ϕ A will always be understood as a function defined on int(K ). In the followings, we state some properties of the functions q A and ϕ A (x), for details see [10].…”

“…By using Corollary 2.2, this contradicts the spherical quasiconvexity of A. In the following two theorems, we present classes of quadratic quasi-convex functions defined in spherically subdual convex sets, which include as particular instances [10,Examples 18 and 19].…”