Clarification in Covering Radius Problem

Question

I am having difficulty understanding this particular definition of Covering Radius Problem - "Given a basis for the lattice, the algorithm must find the largest distance (or in some versions, its approximation) from any vector to the lattice".

Can anyone please explain what is the meaning of "largest distance from any vector to the lattice". How is distance of a vector calculated from a lattice rather than another vector?

Answer 1

If you have a subset $A\subseteq \mathbb R^n$ and a point $x\in \mathbb R^n$ you can define $d(x,A)$ as $\inf\{d(x,a) | a\in A\}$.

Since lattices are closed sets we can consider $\min\{d(x,a)|a\in A\}$