I Radius of the largest ball inside a complex set.

[The Bill]

I've been thinking about notions like the following:
"How far can one be from the nearest road while in a particular country."
"What's the 'maximum thickness' of a subset of \mathbb{R}^n?"
"What mountain range has the biggest circular region entirely within it?"

These sorts of questions lead to defining a quantity which is the "radius of the largest empty (hyper)sphere in the complement of a set" and solving it as a largest empty sphere problem.

Is there a more convenient name for this quantity?

 

[Number Nine]

Do you mean the diameter of a set?
https://en.wikipedia.org/wiki/Diameter#Generalizations

 

[The Bill]

Do you mean the diameter of a set?
https://en.wikipedia.org/wiki/Diameter#Generalizations

No.

sup { d(x, y) | x, y ∈ A } isn't necessarily going to be the same as twice the radius of the largest empty ball in the complement of A.

For example, consider A as a filled in decagram (10 pointed star.) The diameter will be the same as the diameter of its circumcircle, but the quantity that is twice the radius of the largest empty disc in the complement of A will be a fair bit smaller. Exactly how much smaller depends on which type of decagram it is, but you see the point. The inward pointing wedges of empty space between the points of the star limit the size of disk which can fit fully within the decagram.

 

[mfb]

Incircle or inscribed circle in the special case of a triangle. I don't know if there is a name for the general problem.