This is an answer to the general question, it shows that the fugitive can win the game with $\delta_N=\delta_3^{N/2}$, and it also give the best $c$ that is definded in the nice previous answer by Pace Nielsen.

[as I did an edit some definitions might not be optimal anymore, and everything can certainly  be said quicker, I will make it shorter in a couple of days, but not right now, because I have been notified that edits can affect the reading of the thread]

$\forall A,B\subset \mathbb R^2$, $d(A,B):=\inf(\left\{||a-b||, a\in A, b\in B\right\})$
And $Conv(A)$ is the convex hull of $A$.

$Fr(A)$ the frontier of $A$. 

Let $N>0$ be an integer. An **officer strategy** is any $x:\mathbb R^2\mapsto \mathbb (\mathbb R^2)^N$ , $v\mapsto (x_1(v),...,x_N(v))$   (continuous) fonction such that  $\forall u,v\in \mathbb R^2$, $\mathcal D_N(u,v):=||x_1(u)-x_1(v)||+...+||x_K(u)-x_N(v)||\leq ||u-v||$.

> Claim : there exists $\delta\in \mathbb R_+$  such that for all officer strategy $x=(x_1,...,x_n)$,  $\exists f: \mathbb R_+\to \mathbb R^2$ that is 1-lipschitzian, such that $\forall t\in \mathbb R_+ \forall i\in [N]:=\left\{1,...,N\right\}$$||f(t)-x_i\circ f(t)||=:D(t)\geq D(0)\delta$


We fix $N$ and $x$ such that $\forall i\in [N], ||x_i(0)||\geq 1$ and  pose $X_f=X=x\circ f= t\mapsto (X_1(t),...,X_N(t))$.
And also  $X^* = \left\{X_1,...,X_N\right\}$ and $X^*(t)=\left\{X_1(t),...,X_N(t)\right\}$.

It is more convenient to use the continuous paths than alternatives moves with laps etc... but one can easily be convinced that this statement implies the existence of a wining strategy for the fugitive with a $\delta$ that does not depend on the initial configuration. 

As it was mentioned in comments, the fugitive (F) will be safe if he is out of $Conv(X^*)$ the convex hull of the set of officer, we can then procede gradually and show that, there is some $\delta'$ such that the fugitive can escape any convex hull of a choosen subset  $ Y^*\subset X^*$, and then escape a bigger subset $Y^*\cup Y'^*\subset X^*$ (staying at no more then  $\delta'^2$ from everyone)  **with the condition that F will never return in** $conv(Y^*)$ anymore. (This is what we will call the $Y$-game).
Then if in $P$ steps of these kind that will take  F gradually out 
 of $conv(X)$, we can deduce that he was able to escape with $\delta'^P$, hence $\delta'^ N$ can  be took for our general $\delta$ (and even $\delta'^{(N/2)}$, because one can chose  $Y$ to be of maximal cardinal (hense more than $N/2$). Then, it ​is sufficiant  to prove that he can win the $Y$-game : i.e. :

> $\exists \delta'\in \mathbb R_+$, $\forall Y^*\subset X^*$, $0\notin Conv(Y)$, $\exists t'\in \mathbb R_+$,  and $f:\mathbb R_+\to \mathbb R^2$, 1-lipsichian such that $f(0)=0$ and  $\forall t\leq t',f(t)\notin Conv(Y^*(t)))$ and $\exists z\in X^*\setminus Y^*$, $f(t')\notin Conv(Y^*(t')\cup \left\{z(t')\right\})$



We will suppose wlog $Y^*$ maximal (for inclusion), i.e . that $f(0)=0\in \bigcap_{z\in X^*\setminus Y^*} Conv(Y^*\cup \left\{z\right\}):=S_Y$ eitherways we have nothing to do and can take $t'=0$.  We want $f$ to escape $S_Y$ without even enter in $Conv(Y^*)$. This mean he has  to cross a "tangent" to the convex hull of the set of the "back officers" $Y^*$, that passes through some "front officer" in $Z=X\setminus Y$. (the convex hull is not smouth so it is not a tangent, but I'll give precize definitions later). The two nearest "tangent" will be the "target" of the fugitives. They delimit (together with $Fr(conv(Y^*))$ the "cone like"  $S_Y$ that the fugitive is **in** and  that vertex $s_Y$ could be a target too, but there might be a real officer at this place... Note that the fugitive cannot be caught if this vertex is "officer free" and that every officer is into the symetric of the target cone, so it is quite intuitive that it should not be very difficult for the fugitive to touch these segments before getting cought, but we nead a bit more : to prove that he can escape at a distance of more than some $\delta'$ that does not depend on $X$.



Let's then set precizally the definitions that where anounced.


For all $a\in \mathbb R^2$, $A,B\subset \mathbb R^2$

$Tan_A(a)=Fr(Conv(A\cup \left\{a\right\}))\setminus Fr(Conv(A))$ 

$Tan_A(B)=\bigcup_{b\in B} Tan_A(b)$

There exists $s_A(B)\in \mathbb R^2$ such that $Conv(A\cup \left\{s_A(B)\right\})=\bigcap_{b\in B}Conv(A\cup \left\{b\right\}):=S_A(B)$  we call $target_A(B):=tan_A(s_A(B))=Fr(S_A(B))\setminus (Conv(A))$

By now I will  write "$f$ at time $t$,etc." instead of $f(t), etc...$ or just $f, etc...$ when the context is sufficient to dissipate any ambiguity.


If $y\in Y^*$, let's call  $\lambda_y$ the line that join $f$ to $y$. And let $D_s$ be the circle of center $f$ and radius $||f-s||$. Let $y_s=D_s\cap \lambda_y$.
We can wlog suppose that $s$ is an officer : indeed the speed of $s$ cannot be more than the speed of any officer (equility iff $s$ is an officer). Same statement for any $y_s$. Thus the only case we have to study is that of a circle that unit is $1$ and such that there is a semi-circle (that diametre is say $[ab]$ ) on it that contain only $s$ and no other officer. If $f$ is doing any move in the direction of the middle of say $[as]$ that mesure is at most $2^{1/2}$ (unless it is the case for $[bs]$) then it is easy to see that there exists $\delta'$ such that $f$ at full speed is able to  reach the middle at more than $\delta'$ from $s,a$, thus from any officer.

Note that the same argument is still true if we let the sum of officer speed be $(2-\epsilon)\delta$ while a single officer speed is still bounded by $\delta$. Thus the constant $c$ definded in the firts answer cannot be improved.