# On Some Generalized Polyhedral Convex Constructions

## TL;DR

This paper thoroughly studies generalized polyhedral convex sets and functions in locally convex spaces, characterizing their structure and properties, and demonstrating how these concepts apply to scalar optimization problems.

## Contribution

It introduces a characterization of generalized polyhedral convex sets via face finiteness and proves the infimal convolution of certain functions remains polyhedral convex.

## Key findings

- Generalized polyhedral convex sets are characterized by finite faces.
- Infimal convolution of a generalized and a polyhedral convex function is polyhedral.
- Results apply to scalar optimization problems with these structures.

## Abstract

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential, are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized via the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.