A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively. 

 - In a fugitive move, the fugitive can travel no more than a distance of $\delta$
 - In an officer move, the sum of distances travelled by all officers can be no more than $\delta$

Is it true that for $\forall N$, $\exists \delta$ such that the fugitive can escape regardless of the officers' initial distribution? 

Note: If distance between the fugitive and an officer is $0$ in finite moves, the fugitive is **caught**, otherwise they **escape**. I strongly suspect the fugitive can escape if $\delta$ is small enough, but am unable to give a proof. I created this problem myself and know no other existing sources.


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Timothy mentioned the angel problem and fox games in the comments, both of which are discrete games. Let me try to reformulate the problem as a chessboard game while (hopefully) preserving its original flavor as much as possible. We only need to change the second rule to 

 - In an officer move, **only one** officer can move, and they can travel no more than a distance of $\delta$

Notice that if $\delta$ is very small, it is conceivable this revised rule won't substantially change the players strategies.

With this change we can formulate a **dual problem**, in which the fugitive and police officers can move no more than a distance of 1 in their moves, and the nearest officer is at a distance of $D$ away, and ask if for $\forall N$, $\exists D$ such that the fugitive can escape regardless of the officers' initial distribution.

Now we're ready to move the game to the (infinite!) chessboard by imagining the fugitive and the officers moving as **kings** (i.e. each move, whether in the four cardinal directions or diagonally, is of distance 1). The discrete version of the original problem then is this:

**Given a single white king and $N$ black kings, the nearest of which at distance $D$ away, can the white always force a draw without capturing any black kings for some $D$?**

Maybe the above question has an ready answer. If not, I hope an analysis may shed some light for the continuous version.