I would like to prove that for any family of balls $\{B(c\_i,r\_i)\}\_i \subset \mathbb{R}^d$ such that $\{c\_1, \dots, c\_n\} \subset \bigcap\_i B(c\_i,r\_i) $ and $\forall i, r\_i \geq 1$, there exists a ball of radius $1-\frac{\theta\_d}{2}$ included in the intersection $\bigcap\_i B(c\_i,r\_i)$
where $\theta\_d/2 = \frac{1}{2}\sqrt{\frac{2d}{d+1}}$ denotes the ratio between the diameter and the radius of the smallest enclosing ball of a regular simplex. 


Intuitively, it seems that the way of making the smallest intersection is to assign all points $c\_i$ to the vertices of a regular simplex of diameter $1$ and all $r\_i$ to $1$. By doing so, one can check that the ball of radius $1-\frac{\theta\_d}{2}$ centered at $x$ the barycenter of $\{c\_1,\dots,c\_n\}$ is included in the intersection of balls $ \bigcap\_i B(c\_i,r\_i)$.
Indeed, in the case of a simplex, the radius of the biggest ball centered at $x$ and included in the intersection of balls is $1-\text{Radius}(\sigma) = 1 - \frac{\theta\_d}{2}$ (hence the constant is tight in this case).


I am having difficulties to prove that this case is indeed the worst case. 
I was just able to prove that the result holds when all balls have the same radius.
Does this result seems familiar to someone? I would really appreciate any comment, idea or reference.


ps : The topology tag is here for several reason. One of them is that the biggest radius of the ball included in the intersection corresponds to the weak feature size of the complement of the intersection. Another one is that this result is linked to a collapsibily result.



Biggest ball included in an intersection of balls
OK, let's try.


First, $n=d+1$ (Helly's theorem).


Second, if the balls of radii $r\_i-\theta\_d$ do not have a common point, there is $\theta<\theta\_d$ such that the balls of radii $r\_i-\theta$ have exactly one common point, say, the origin. Also, we can ignore the balls such that $0$ is not on their boundary. Thus, we get $m\le d+1$ vectors $v\_i$ of length $r\_i-\theta$ with $|v\_i-v\_j|\le \min(r\_i,r\_j)$. Moreover, the vectors $v\_i$ cannot lie in one half-space (otherwise there would be more common points), so $\sum\_i a\_iv\_i=0$ for some $a\_i\ge 0$, not all of them $0$. 


Now just note that as soon as the angle between $v\_i$ and $v\_j$ is obtuse, decreasing each length keeping $|v\_i-v\_j|$ constant will increase the angle and that in the case of the equilateral triangle, setting $r\_i=1$ will also increase the angle. Thus, if $e\_i$ are unit vectors in the directions of $v\_i$, we have $\langle e\_i,e\_j\rangle> -1/d$ and we still have $\sum b\_i e\_i=0$ with $b\_j\ge 0$. However, the matrix with $1$'s on the diagonal and off-diagonal terms satisfying $0>A\_{ij}>-1/d$ is positive definite. We thus run into a contradiction.


I hope this holds water. Check everything and let me know if there are any gaps. :) 


