from $p_{C_1}$ to any $q\in C_2$ is at most $2\operatorname{opt}_k+R$. We conclude that merging $C_1$ and $C_2$ would result in a cluster whose discrete radius can be upper bounded by $\operatorname{drad}(C_1\cup C_2)<\operatorname{cost_{drad}}({\mathcal{C}}_m)+2\operatorname{opt}_k$ (see Figure \[drad\_merge\_fig\]). The result follows using $C_1,C_2\in{\mathcal{C}}_{t+1}$ and Observation \[drad\_greedy\_obs\].

To prove Theorem \[drad\_result\], we apply Lemma \[drad\_rem\_phaselem\] for about $\log k$ consecutive phases.

Let $\varepsilon>0$ and $u:=\left\lceil\log_2(k)+\varepsilon\right\rceil$ such that $\log_2k<u\leq\log_2(k)+1$. Furthermore, define $m_i:=k+\left\lfloor\left(\frac{1}{2}\right)^ik\right\rfloor$ for all $i=0,\ldots,u$. Then, $m_u=k$ and $m_i>k$ for all $i=0,\ldots,u-1$. Since $k+\left\lfloor\frac{m_i-k}{2}\right\rfloor=k+\left\lfloor\frac{1}{2}\left\lfloor\left(\frac{1}{2}\right)^ik\right\rfloor\right\rfloor\leq k+\left\lfloor\left(\frac{1}{2}\right)^{i+1}k\right\rfloor=m_{i+1}$ and Algorithm \[drad\_algo\] uses a greedy strategy, we deduce $\operatorname{cost_{drad}}({\mathcal{C}}_{m_{i+1}})\leq\operatorname{cost_{drad}}\left({\mathcal{C}}_{k+\left\lfloor\frac{m_i-k}{2}\right\rfloor}\right)$ for all $i=0,\ldots,u-1$. Combining this with Lemma \[drad\_rem\_phaselem\] (applied to $m=m_i$), we obtain $$\operatorname{cost_{diam}}({\mathcal{C}}_{m_{i+1}})<\operatorname{cost_{drad}}({\mathcal{C}}_{m_i})+2\operatorname{opt}_k.$$ By repeatedly applying this inequality for $i=0,\ldots,u-1$ and using $m_0=2k$, we get $$\operatorname{cost_{drad}}({\mathcal{C}}_k)<\operatorname{cost_{drad}}({\mathcal{C}}_{2k})+\sum_{i=0}^{u-1}2\operatorname{opt}_k\leq\operatorname{cost_{drad}}({\mathcal{C}}_{2k})+(2\log_2(k)+2)\cdot\operatorname{opt}_k.$$ Hence, the result follows using Proposition \[drad\_2k\_prop\].

$k$-center clustering {#rad_sec}
---------------------

The agglomerative algorithm for the $k$-center problem is stated as Algorithm \[rad\_algo\].

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<span style="font-variant:small-caps;">AgglomerativeRadius</span>$(X)$:\

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    $X$ finite set of input points from ${\mathbb{R}}^d$
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  : The agglomerative algorithm for the $k$-center problem.[]{data-label="rad_algo"}

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${\mathcal{C}}_{|X|}:=\left\{\,\{x\}\,|\,x\in X\right\}$ find distinct clusters $A,B\in {\mathcal{C}}_{i+1}$ minimizing $\operatorname{rad}(A\cup B)$\[rad\_algo\_min\] ${\mathcal{C}}_i:=({\mathcal{C}}_{i+1}\setminus\{A,B\})\,\cup\,\{A\cup B\}$ ${\mathcal{C}}_1,\ldots,{\mathcal{C}}_{|X|}$

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The only difference to Algorithm \[drad\_algo\] is the minimization of the radius instead of the discrete radius in Step \[rad\_algo\_min\].

In the 