Conditionally Evenly Convex Sets and Evenly Quasi-Convex Maps

TL;DR

This paper generalizes evenly convex sets and evenly quasi-convex maps within a conditional framework, extending classical theorems and providing dual representations relevant to topological vector spaces.

Contribution

It introduces a conditional version of evenly convex sets and extends the bipolar theorem, leading to new dual representations of conditionally evenly quasi-convex maps.

Findings

Generalized evenly convex sets in a conditional setting.

Established a conditional bipolar theorem.

Derived dual representations for conditionally evenly quasi-convex maps.

Abstract

Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem. This notion is then applied to obtain the dual representation of conditionally evenly quasi-convex maps.