Can the fugitive escape?

Question

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 d.
• In an officer move, the sum of distances travelled by all officers can be no more than d.

The fugitive is caught if their distance to some officer is 0 in finite moves, otherwise they escape.

Question: Given N, is there always some $d \gt 0$ for which the fugitive can escape, regardless of the officers' initial distribution?

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Related: One king vs many. Can white force a draw? (discrete version dual problem)

Answer 1

Partial answer (a crude upper bound):

for sufficiently large $N$ (at least, it works for magnitudes of about $20$), if such $d$ exists, it must be $$d<\frac{2}{\sqrt3}\sin\frac\pi N$$

Because

Assume that the officers stand on the unit circle in the vertices of a regular $N$-gon inscribed into that circle. The distance between the officers than will be $2 \sin\frac\pi N$, the side length of such a polygon.
On the other side, if the distance between officers is $d\sqrt3$ or less (i.e. when $d$ is too large), then the common chord of the 2 circles with radii $d$ centered on the adjacent officers will have length of $d$ or greater, so the fugitive cannot pass inbetween these officers to cross the boundary. So, $2\sin\frac\pi N > d\sqrt3$, and we have the bound for $d$ written above.
OF course, the $d>1$ case won't work for sufficiently large value of $N$, since after making the first move, the fugitive will be at a distance of $d-1$ from the circle, and $d-1+\varepsilon$ from the closest officer, where $\varepsilon$ is much less then 1 for large $N$, so the fugitive will be caught next move.

Answer 2

No, the fugitive cannot escape with large N

Assume an arbitrarily large number of officers surrounding the fugitive, all standing 1 unit away from the fugitive on the perimeter of the unit circle centered on the fugitive. The fugitive cannot end his move within distance d of any officer, or that officer can catch him next turn.

In the limit of an infinite number of officers, the "disallowed" region for the fugitive is an annulus with width 2d. The fugitive cannot cross a width of 2d in a single move if his maximum move length is d. If d > 1, the "disallowed" region is a circle of radius d+1, which also cannot be escaped with a single move of length d.

NOTE: This reasoning is incorrect, as you can always pick d smaller than half the distance between officers, so that the "disallowed" region is just a series of circles and not an annulus. There can always be a viable escape path in the initial configuration, so long as d is small enough.