Title: Interval-type theorems concerning means,
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.