Let $L$ be an $n-$dimensional lattice. The Voronoi region of $L$ is given by
$$
\mathcal{V}(L)=\big\{x\in\mathbb{R}^n~|~ \|x\|\_2\leq \|x-v\|\_2~\forall v\in L\setminus\{0\}\big\}.
$$
Considering the half-spaces
$$
H\_v=\left\{x\in\mathbb{R}^n~\Big|~\langle x,v \rangle\leq \frac{1}{2}\|v\|\_2^2\right\}
$$
the normal vector of the halfspace $H\_v$ is $v$ and it contains the point $v/2$ on its boundary. We can write $\mathcal{V}$ as
$$
\mathcal{V}=\bigcap\_{v\in L\backslash\{0\}}H\_v
$$
Every irredundant halfspace $H\_v$ induces an $(n−1)$-dimensional face of $\mathcal{V}$. The set of such $H\_v$ is finite and their related vectors $v$ are said to be relevant.
Clearly, the covering radius $r(L)$ of $L$ is the length of the vertex of (the polytope) $\mathcal{V}$ with maximal norm.


**My question is**: what is the most efficient way to get $r(L)$ given the complete set of relevant vectors?



Covering radius of a lattice from relevant vectors
A convex polyhedra has two (equivalent) representations:


1. The *H-representation*, as an intersection of finitely many halfspaces.
2. The *V-representation*, as the convex hull of finitely many vertices.


The problem of converting between these is the problem of "Polyhedral Representation Conversion". In this language, it seems that you want to convert an $H$-representation polyhedra to a $V$-representation polyhedra, where you can then compute the norm of each vertex.
See [chapter 9 of these notes](http://www-oldurls.inf.ethz.ch/personal/fukudak/lect/pclect/notes2015/PolyComp2015.pdf) for more details.


If by "efficient" you mean in real life, there are various programs for computations involving convex polyhedra which can do this computation. The tool [polymake](https://www.polymake.org/doku.php) is probably the best known, but the computer algebra system [Sage](https://doc.sagemath.org/html/en/thematic_tutorials/geometry/polyhedra_tutorial.html#lecture-1-representation-objects) can do the computation as well.


