A note on $p^\lambda$-convex set in a complete Riemannian manifold

TL;DR

This paper generalizes the concept of $ ext{p}^ ext{lambda}$-convex sets in complete Riemannian manifolds, explores their properties, and clarifies the relationship with geodesic convexity, including counterexamples and conditions for equivalence.

Contribution

It introduces the notion of $p^ ext{lambda}$-convex functions in Riemannian manifolds and analyzes their relation to geodesic convex sets, extending existing convexity concepts.

Findings

Counterexample shows $ ext{lambda}$-radial contraction of a geodesic need not be a geodesic.

Under certain conditions, $ ext{p}^ ext{lambda}$-convex sets are equivalent to geodesic convex sets.

The paper establishes relations between $ ext{p}^ ext{lambda}$-convexity and geodesic convexity.

Abstract

In this paper we have generalized the notion of $\lambda$-radial contraction in complete Riemannian manifold and developed the concept of $p^\lambda$-convex function. We have also given a counter example proving the fact that in general $\lambda$-radial contraction of a geodesic is not necessarily a geodesic. We have also deduced some relations between geodesic convex sets and $p^\lambda$-convex sets and showed that under certain conditions they are equivalent.