What does majorization mean in the context of intervals?

Question

I just got this book on convex optimization, and in the preliminary section they show syntax for "majorized" and "minorized" intervals as

I searched the terms majorized and minorized within math.stackexchange and elsewhere on the internet, but could not find anything related to intervals.

The closest comes from wikipedia:

For a vector $ \mathbf {a} \in \mathbb {R} ^{d}$ we denote by $ \mathbf {a} ^{\downarrow }\in \mathbb {R} ^{d} $ the vector with the same components, but sorted in descending order. Given $ \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{d} $, we say that $ \mathbf {a} $ weakly majorizes (or dominates) $ \mathbf {b} $ from below written as $ \mathbf {a} \succ _{w}\mathbf {b} $ iff

$ \sum _{i=1}^{k}a_{i}^{\downarrow }\geq \sum _{i=1}^{k}b_{i}^{\downarrow }\quad {\text{for }}k=1,\dots ,d $

But I still have two questions:

1. The order of a sequence doesn't affect its sum, so why is it significant to use the ordered vectors $ \mathbf{a}^{\downarrow} $ and $ \mathbf{b}^{\downarrow} $ within the sums?

2. How does that inequality relate to intervals?