Say, that you have a N-dimensional hypercube with the edge of 1 (unit).
You can then, of course, put 2^N spheres inside, each with a diameter of 1/2.
For example, eight 50 cm diameter spheres, inside a 1 meter cube, fit nicely. As four circles each of 50 cm diameter fit well inside a unit square.
Now, you squeeze another sphere in the middle, touching every other of those 2^N spheres.
It has been recently noted, that the middle sphere can have a diameter bigger than 1!
Question No. 1: At which dimension X, does this (first) happen?
Question No. 2: At which dimension Y (if at any), the middle hypersphere has bigger hypervolume than that of its “encompassing” hypercube?
Question No. 3: How many 100 dimensional hyperspheres with the diameter of 2, can be squeezed (with no deformation) inside a 100 dimensional hypercube with the edge of 1?