Help in reading Set Notation

Question

I am reading the paper on optimization and the following is the notation: I am trying to understand how to read it because there is equality and inquality in the same equation. I know the norm term but i dont understnd what this equation as a whole is read as.

Answer 1

That's not an equation or an inequation: that is just the set-bulder notation. Specifically, the line reads:

$B_{r_j}(x_j)$ is the set of all the elements $x$ of the set $\mathcal X$ which satisfy $\lVert x-x_j\rVert\le r_j$.

In oter words: $B_{r_j}(x_j)$ is the set of the elements of $\mathcal X$ which are in closed ball with center $x_j$ and radius $r_j$ with respect to the norm $\lVert \bullet\rVert$.

Answer 2

I would read this as "The set $B_{r_j}(x_j)$ is equal to the set of all $x$ in the set $\chi$ such that $|| x_j - x|| \leq r_j$."