Monotonicity of sets in Hadamard spaces from polarity point of view

TL;DR

This paper explores the properties of monotone sets in Hadamard spaces, introduces flat Hadamard spaces, and characterizes maximal monotone sets using polarity, revealing their closure properties.

Contribution

It introduces the notion of monotone sets in Hadamard spaces, characterizes flat spaces via the $ ext{F}_l$-property, and provides polarity-based characterizations of maximal monotone sets.

Findings

Flat Hadamard spaces characterized by $ ext{F}_l$-property.

Maximal monotone sets are sequentially closed in a specific topology.

Polarity provides a new perspective on monotonicity in Hadamard spaces.

Abstract

This paper is devoted to introduce and investigate the notion of monotone sets in Hadamard spaces. First, flat Hadamard spaces are introduced and investigated. It is shown that an Hadamard space $X$ is flat if and only if $X\times X^\medlozenge$ has $\mathcal{F}_l$-property, where $X^\medlozenge$ is the linear dual of $X$. Moreover, monotone and maximal monotone sets are introduced and also monotonicity from polarity point of view is considered. Some characterizations of (maximal) monotone sets, specially based on polarity, are given. Finally, it is proved that any maximal monotone set is sequentially $bw\times${$\|\cdot\|_\loz$}-closed in $X\times X^\medlozenge$.