for different values of $k$ hierarchical, if it fulfills the following two properties. First, for any $1\leq k\leq|X|$ the collection contains at most one $k$-clustering. Second, for any two of its clusterings ${\mathcal{C}}_i,{\mathcal{C}}_j$ with $|{\mathcal{C}}_i|=i<j=|{\mathcal{C}}_j|$, every cluster in ${\mathcal{C}}_i$ is the union of one or more clusters from ${\mathcal{C}}_j$. A hierarchical collection of clusterings is called a hierarchical clustering.

We define the diameter of a finite and non-empty set $C\subset{\mathbb{R}}^d$ to be $\operatorname{diam}(C):=\max_{x,y\in C}\|x-y\|$. Furthermore, we define the diameter cost of a $k$-clustering ${\mathcal{C}}_k$ as its largest diameter, i.e. $\operatorname{cost_{diam}}({\mathcal{C}}_k):=\max_{C\in {\mathcal{C}}_k}\operatorname{diam}(C)$. The radius of $C$ is defined as $\operatorname{rad}(C):=\min_{y\in{\mathbb{R}}^d}\max_{x\in C}\|x-y\|$ and the radius cost of a $k$-clustering ${\mathcal{C}}_k$ is defined as its largest radius, i.e. $\operatorname{cost_{rad}}({\mathcal{C}}_k):=\max_{C\in {\mathcal{C}}_k}\operatorname{rad}(C)$. Finally, we define the discrete radius of $C$ to be $\operatorname{drad}(C):=\min_{y\in C}\max_{x\in C}\|x-y\|$ and the discrete radius cost of a $k$-clustering ${\mathcal{C}}_k$ is defined as its largest discrete radius, i.e. $\operatorname{cost_{drad}}({\mathcal{C}}_k):=\max_{C\in {\mathcal{C}}_k}\operatorname{drad}(C)$.

\[drad\_prob\] Given $k\in{\mathbb{N}}$ and a finite set $X\subset{\mathbb{R}}^d$ with $|X|\geq k$, find a $k$-clustering ${\mathcal{C}}_k$ of $X$ with minimal discrete radius cost.

\[rad\_prob\] Given $k\in{\mathbb{N}}$ and a finite set $X\subset{\mathbb{R}}^d$ with $|X|\geq k$, find a $k$-clustering ${\mathcal{C}}_k$ of $X$ with minimal radius cost.

\[diam\_prob\] Given $k\in{\mathbb{N}}$ and a finite set $X\subset{\mathbb{R}}^d$ 