Not an answer; just an illustration.
I had some difficulty understanding the question, so...

Here $n=2$, so the shapes are planar, $\mathbb{R}^2$.
I used Reuleaux triangles for the $3=n{+}1$ unit-constant-width bodies
$F=\{ B_1, B_2, B_3 \}$ forming an "impressive" and "assuming" family $F$.
A particular point $x \in \mathbb{R}^2$ is shown, with segments achieving
$d(x,B_i)$. In this case, all three of those min-distances to the
bodies are equal, so that is also the max $\gamma_2$.
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![Reuleaux][1]
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So I think the question is simply asking if there is a lowerbound
on the radius of a ball that can nestle in the gap.?
I.e., can we ensure that the gap is not arbitrarily small?

Apologies if I am misinterpreting...

**Added**: To address $d{=}3$ & Per A.'s question, here is an image
of a constant-width [Reuleaux tetrahedron](http://en.wikipedia.org/wiki/Reuleaux_tetrahedron):
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<img src="https://i.sstatic.net/3FfLX.png" width="300" />


  [1]: https://i.sstatic.net/GM5Cp.jpg