Bernstein--Doetsch type theorems with Tabor type error terms for set-valued maps

TL;DR

This paper extends classical convexity theorems to set-valued maps, incorporating Tabor type error terms, and explores their Jensen convexity and concavity properties.

Contribution

It introduces Bernstein--Doetsch type theorems with Tabor error terms for set-valued maps, advancing the understanding of approximate convexity.

Findings

Established Bernstein--Doetsch type theorems with Tabor error terms

Characterized strongly and approximately Jensen convex and concave set-valued maps

Provided new insights into the structure of approximate convexity for set-valued functions

Abstract

In this paper, we investigate set-valued maps of strongly and approximately Jensen convex and Jensen concave type. We present counterparts of the Bernstein--Doetsch Theorem with Tabor type error terms.