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Definition 0. Given a metric space $X$ and subsets $H$ and $S$ thereof, define: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$

(This as an asymmetric version of the Hausdorff distance.)

Here's some slightly dodgy intuition about why this definition is interesting: imagine that $S$ is a set of "safety points" and $H$ is the set of "homes where people live." If a disaster strikes and people begin making their way to the nearest safety point, then $d(H,S)$ (read: distance from houses to safety) describes the elapsed time until everyone gets to safety, assuming that everyone leaves immediately and moves with speed $1$.

I'm interested in the following problem: $H$ = "the homes where people live" is given, and we need to place $k$-many safety points in an optimal way, so that the elapsed time before everyone is safe is minimized. More formally:

Problem. Given

  • $H$, a bounded subset of $\mathbb{R}^n$, and
  • $k$, a natural number

find:

  • a subset $S$ of $\mathbb{R}^n$ satisfying $|S| \leq k$, such that for all "competing" subsets $S'$ of $\mathbb{R}^n$ satisfying $|S'| \leq k,$ we have $d(H,S) \leq d(H,S')$.

Question. Have any tools or techniques been developed to solve this problem for sufficiently nice $H$?

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goblin GONE
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  • Nice question! Your question, I think, is similar (but is not the same!) to the "optimal ball backing problem", where given a solid $H$ you are looking for the largest number $k$ of pairwise disjoint balls of fixed radius (say, $1/2$) contained in $H$. Given the set $S$ of centers of such collection of balls, we have that the balls $B(s,1), s\in S$, cover $H$ which implies that $d(H,S)\le 1$. I think both questions are equally hard and it makes sense for you to check the ball packing literature to see if some ideas are useful in your setting. – Moishe Kohan May 30 '16 at 08:46
  • @studiosus, thanks for the tip. (And the kind words.) – goblin GONE May 30 '16 at 08:57

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