# A convex extension of lower semicontinuous functions defined on normal Hausdorff space

## TL;DR

This paper demonstrates that minimization problems involving proper lower semicontinuous functions on normal Hausdorff spaces can be transformed into equivalent convex minimization problems in dual Banach spaces, facilitating analysis and solution.

## Contribution

It introduces a canonical transformation linking lower semicontinuous functions on normal Hausdorff spaces to convex functions on dual Banach spaces, preserving minimization problems.

## Key findings

- Establishes a bijective operator between the two classes of functions.
- Proves the equivalence of minimization problems in different settings.
- Shows existence of a convex extension for lower semicontinuous functions.

## Abstract

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined on a weak * convex compact subset of some dual Banach space. We estalish the existence of an bijective operator between the two classes of functions which preserves the problems of minimization.