Perspective Functions: Properties, Constructions, and Examples

TL;DR

This paper explores the properties and construction methods of perspective functions, highlighting their applications in convex analysis and providing new examples and general schemes for their use.

Contribution

It offers a comprehensive analysis of perspective functions' properties and introduces novel construction schemes and examples, expanding their applicability in convex analysis.

Findings

Analysis of algebraic and convex-analytical properties

Development of general schemes for constructing convex functions

Presentation of new examples of perspective functions

Abstract

Many functions encountered in applied mathematics and in statistical data analysis can be expressed in terms of perspective functions. One of the earliest examples is the Fisher information, which appeared in statistics in the 1920s. We analyze various algebraic and convex-analytical properties of perspective functions and provide general schemes to construct lower semicontinuous convex functions from them. Several new examples are presented and existing instances are featured as special cases.