# Interval-type theorems concerning means

## TL;DR

This paper explores the structure of families of means under a natural order, introducing the concept of interval-type sets, and investigates these properties specifically for Gini and Hardy means, also discussing metrics among means.

## Contribution

It introduces the notion of interval-type sets within families of means and analyzes their properties for Gini and Hardy means, including metric considerations.

## Key findings

- Interval-type sets correspond to real intervals in power means.
- Certain properties of interval-type sets are identified for Gini and Hardy means.
- Results on the $L^0$ metric among abstract means are provided.

## Abstract

Each family $\mathcal{M}$ of means has a natural, partial order (point-wise order), that is $M \le N$ iff $M(x) \le N(x)$ for all admissible $x$. In this setting we can introduce the notion of interval-type set (a subset $\mathcal{I} \subset \mathcal{M}$ such that whenever $M \le P \le N$ for some $M,\,N \in \mathcal{I}$ and $P \in \mathcal{M}$ then $P \in \mathcal{I}$). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. In the present paper we consider this property for Gini means and Hardy means. Moreover some results concerning $L^\infty$ metric among (abstract) means will be obtained.