Let $\mathbb{R}^n$ be a normed affine space of finite dimension $n$, and $d : \mathbb{R}^n \times \mathbb{R}^n \mapsto \mathbb{R}^+$ be the distance derived from the norm under consideration. A convex polytope $P \subseteq \mathbb{R}^n$ is the intersection of any finite number of closed half-spaces from $\mathbb{R}^n$, such that $P \neq \emptyset$. $Pol_n$ denotes the set of all convex polytopes from $\mathbb{R}^n$. A profile $\mathcal{P}$ is a non-empty multiset $\{P_1, \dots, P_k\}$, $k > 0$, of convex polytopes from $Pol_n$. $\mathcal{Pol}_n$ denotes the set of all profiles. $d$ is extended to the mapping $d : \mathbb{R}^n \times Pol_n \mapsto \mathbb{R}^+$ defined $\forall p \in \mathbb{R}^n$, $\forall P \in Pol_n$ as $d(p, P) = \min\{d(p, p') \mid p' \in P\}$. Then, $d$ is extended to the mapping $d : \mathbb{R}^n \times \mathcal{Pol}_n \mapsto \mathbb{R}^+$, defined $\forall p \in \mathbb{R}^n$, $\forall \mathcal{P} \in \mathcal{Pol}_n$ as $d(p, \mathcal{P}) = \sum\{d(p, P) \mid P \in \mathcal{P}\}$.

I would like to show that for any profile $\mathcal{P}$, the set $argmin\{d(p, \mathcal{P}) \mid p \in \mathbb{R}^n\}$ (i.e., the set $\{p \in \mathbb{R}^n \mid \forall p' \in \mathbb{R}^n, d(p, \mathcal{P}) \leq d(p', \mathcal{P})\}$) is a convex polytope (or to find a counter-example, that I failed to do so far).

argminexpression. Sorry! $\endgroup$ – Joseph O'Rourke May 26 '14 at 18:22