Interval-type theorems concerning means

Each family ℳ of means has a natural, partial order (point-wise order), that is M ≤ N iff M(x) ≤ N(x) for all admissible x.

In this setting we can introduce the notion of interval-type set (a subset ℐ ⊂ℳ such that whenever M ≤ P ≤ N for some M, N ∈ℐ and P ∈ℳ then P ∈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^∞ metric among (abstract) means will be obtained.