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?

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.