Let $X=\{x_{1},x_{2},\dots,x_{k}\}\subset R^2, \hspace{2pt} k>2$. The smallest covering radius of X is defined as
$$r^*(X)=argmin\{r:\exists x\in R^2 \hspace{2pt} \text{such that}\hspace{2pt} \|x_{j}-x\|\leq r \hspace{4pt}\text{for all} \hspace{3pt} j= 1,2,\dots,k\}$$ Using Helly's Theorem I want to compute the smallest covering radius for X efficiently, i.e. with computational complexity that grows polynomially in k. Also does the covering radius depend upon the norm used? (Here the definition involves $l_{2}$-norm)