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Offball: object whose projections are disks

Q: ??? An "offball" is an object such that its projections in three orthogonal directions are unit disks. What's the max/min volume it can have, assuming it's convex?
[rec.puzzles, 16 Jul 1996]

A: The maximal volume is attained by the intersection of three orthogonal unit cylinders - it's clear no point in the object can be outside this region. This offball looks like an over-inflated rhombic dodecahedron.

A possible offball smaller than the unit sphere is the convex hull of three orthogonal unit disks, which is strictly contained in the unit sphere. However, there are possible offballs not containing this one. For example, take a small wedge of one of the disks far from the other disks and move it out of the plane of its disk slightly, then take the convex hull of the result. Hence it's not clear that the original convex hull has the minimal volume.