Reverses of the Triangle Inequality in Banach Spaces

TL;DR

This paper surveys recent reverses of the triangle inequality in Banach spaces, introduces new results, and discusses specific cases in Hilbert spaces, complex numbers, and functions, enhancing understanding of inequalities in functional analysis.

Contribution

It provides a comprehensive survey of recent triangle inequality reverses and presents new findings, with particular focus on Hilbert spaces and complex-valued functions.

Findings

Survey of recent triangle inequality reverses in Banach spaces

Introduction of new inequality reversal results

Application to Hilbert spaces and complex functions

Abstract

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.