# Quasi-Relative Interiors for Graphs of Convex Set-Valued Mappings

## TL;DR

This paper extends the concept of quasi-relative interior for convex sets, providing new formulas for convex graphs and epigraphs, and demonstrates its analogous role to relative interior in finite-dimensional spaces.

## Contribution

It introduces new formulas for quasi-relative interiors of convex graphs and epigraphs, enhancing understanding in infinite-dimensional convex analysis.

## Key findings

- New formulas for quasi-relative interiors of convex graphs
- Representation of quasi-relative interiors of convex epigraphs
- Similar roles of quasi-relative interior in infinite and finite dimensions

## Abstract

This paper aims at providing further studies of the notion of quasi-relative interior for convex sets introduced by Borwein and Lewis. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings and for convex epigraphs of extended-real-valued functions defined on locally convex topological vector spaces. We also show that the role, which this notion plays in infinite dimensions and the results obtained in this vein, are similar to those involving relative interior in finite-dimensional spaces.