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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 (from here) of a constant-width Meissner tetrahedron:

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![CWidth3D][2]

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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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 (from here) of a constant-width Meissner tetrahedron:

---

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 $\gamma_2$.

---

  ![Reuleaux][1]

---

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 image (from here) of a constant-width Meissner tetrahedron:

---

![CWidth3D][2]

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 $\gamma_2$.

---

  ![Reuleaux][1]

---

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 (from here) of a constant-width Meissner tetrahedron:

---

![CWidth3D][2]

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 $\gamma_2$.

---

  ![Reuleaux][1]

---

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:

---

         

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 $\gamma_2$.

---

  ![Reuleaux][1]

---

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 image (from here) of a constant-width Meissner tetrahedron:

---

![CWidth3D][2]