A fugitive is surrounded by $N$ policemen, with the nearest one at distance $1$ away. They 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 $\forall N$, $\exists \delta$ such that the fugitive can escape? If there's more than one fugitive who can move no more than $\delta$ collectively in their turn, can all of them escape?

I strongly suspect the fugitive can escape (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.