Continued fractions built from convex sets and convex functions

TL;DR

This paper generalizes continued fractions within convex analysis, establishing convergence conditions for various convex structures and transforms, including convex sets and functions, with applications to Minkowski addition and Legendre--Fenchel transforms.

Contribution

It introduces a unified framework for continued fractions in convex analysis and provides convergence criteria for different convex structures and transforms.

Findings

Convergence conditions for continued fractions with deterministic terms.

Necessary and sufficient conditions for convex sets with Minkowski addition.

Convergence criteria for non-negative convex functions with Legendre--Fenchel transforms.

Abstract

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform (where also necessary and sufficient conditions of convergence for continued fractions with constant terms are obtained) and the family of non-negative convex functions with the Legendre--Fenchel and Artstein-Avidan--Milman transforms.