A fugitive is surrounded by $N$ policemen, with the nearest one at distance $1$ away. The fugitive and the policemen take turns to move. The fugitive can move no more than a distance of $\delta$, and the policemen collectively can move no more than a distance of $\delta$, in their turns respectively.

Is it true that for $\forall N$, $\exists \delta$ such that the fugitive can escape regardless of the policemen's initial distribution? If there's more than one fugitive who can move no more than $\delta$ collectively in their turn, can all of them escape?

If distance between the fugitive and a policeman is $0$ in finite moves, the fugitive is caught, otherwise they escape. I strongly suspect the fugitive can escape if $\delta$ is small enough (at least in the one-fugitive case), but am unable to give a proof. I created this problem myself and know no other existing sources.