Geometry:what is the opposite of a sphere?& what shape maximises surface area/volume?

[StephenG1CMZ]

Over on quora I just saw an interesting question: What is the opposite of a sphere?

My non-technical guess was all the space outside the sphere (like a torus but in 3D). That has the advantage of being easily visualisable in 3D.

But what is the defining characteristic of a sphere that we want the opposite of?
Some answers suggested a negative curvature, and that sounds mathematically interesting, but I'm not yet visualising how that looks.
Other answers picked on the number of corners, but I don't think it has a definite number. It seems to me you can make a case for both zero and infinity.

Another answer suggested that a sphere minimises suface area/volume, so its opposite should instead maximise that ratio...But what shape has that characteristic?