Generalized Relative Interiors and Generalized Convexity in Infinite Dimensions

TL;DR

This paper introduces new concepts of generalized relative interiors and quasi-near convexity in infinite-dimensional spaces, extending classical notions to broader set classes and providing new insights into convexity and set-valued mappings.

Contribution

It develops a novel notion of quasi-near convexity and explores its relationship with generalized relative interiors in infinite-dimensional topological vector spaces.

Findings

Introduces quasi-near convexity as an extension of near convexity.

Provides new results on generalized relative interiors for set-valued mappings.

Extends classical convexity notions to infinite-dimensional settings.

Abstract

This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We introduce, study, and utilize a novel notion of quasi-near convexity of sets that is an infinite-dimensional extension of the widely acknowledged notion of near convexity. Quasi-near convexity is associated with the quasi-relative interior of sets, which is investigated in the paper together with other generalized relative interior notions for sets, not necessarily convex. In this way, we obtain new results on generalized relative interiors for graphs of set-valued mappings in convexity and generalized convexity settings.