Generalized Hukuhara directional differentiability of interval-valued functions on Riemannian manifolds

TL;DR

This paper investigates the properties of interval-valued functions on Riemannian manifolds, revealing that generalized Hukuhara directional differentiability does not always align with the differentiability of associated functions, challenging previous assertions.

Contribution

It clarifies the relationship between Hukuhara differentiability and the differentiability of center and half-width functions on Riemannian manifolds, correcting prior misconceptions.

Findings

Differentiability of IVF does not imply differentiability of center and half-width functions.

Contradicts previous claims that endpoint functions' differentiability is equivalent to IVF differentiability.

Provides basic results linking convexity and directional differentiability of IVF.

Abstract

In this paper, we show that generalized Hukuhara directional differentiability of an interval-valued function (IVF) defined on Riemannian manifolds is not equivalent to the directional differentiability of its center and half-width functions and hence not to its end point functions. This contrasts with S.-L. Chen's \cite{chen} assertion which says the equivalence holds in terms of endpoint functions of an IVF which is defined on a Hadamard manifold. Additionally, the paper addresses some other inaccuracies which arise when assuming the convexity of a function at a single point in its domain. In light of these arguments, the paper presents some basic results that relate to both the convexity and directional differentiability of an IVF.