On geodesic strongly E-convex sets and geodesic strongly E-convex functions

Adem Kılıçman,Wedad Saleh

doi:10.1186/s13660-015-0824-z

Abstract

In this article, geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold are extended to the so-called geodesic strongly E-convex sets and geodesic strongly E-convex functions. Some properties of geodesic strongly E-convex sets are also discussed. The results obtained in this article may inspire future research in convex analysis and related optimization fields.

Highlights

Convexity and its generalizations play an important role in optimization theory, convex analysis, Minkowski space, and fractal mathematics [ – ]
2 Preliminaries we introduce some definitions and well-known results of Riemannian manifolds, which help us throughout the article
In, the geodesic E-convex set and geodesic E-convex functions on a Riemannian manifold were introduced by Iqbal et al [ ] as follows

Summary

Introduction

Convexity and its generalizations play an important role in optimization theory, convex analysis, Minkowski space, and fractal mathematics [ – ]. In , the geodesic E-convex set and geodesic E-convex functions on a Riemannian manifold were introduced by Iqbal et al [ ] as follows. A subset B in a Riemannian manifold N is called GSEC if and only if there is a unique geodesic ηαb +E(b ),αb +E(b )(γ ) of length d(b , b ), which belongs to B, ∀b , b ∈ B, α ∈ [ , ], and γ ∈ [ , ].

Results

Conclusion