Non-existence of certain type of convex functions on a Riemannian manifold with a pole

TL;DR

This paper investigates the non-existence of specific convex functions on Riemannian manifolds with a pole, introducing new notions of odd and even functions and deriving related inequalities.

Contribution

It introduces the concepts of odd and even functions on such manifolds and proves the non-existence of certain convex functions with complete gradients.

Findings

Non-existence of non-trivial, non-negative differentiable odd convex functions with complete gradients.

Development of isoperimetric inequalities related to convex functions on manifolds.

Introduction of new notions of odd and even functions on Riemannian manifolds with a pole.

Abstract

This paper is devoted to the study of non-existence of certain type of convex functions on a Riemannian manifold with a pole. To this end, we have developed the notion of odd and even function on a Riemannian manifold with a pole and proved the non-existence of non-trivial and non-negative differentiable odd convex function whose gradient is complete. Finally, we have deduced some isoperimetric type inequality related with convex function.