I’ve been up all night so I can sleep early tonight, wake up really early tomorrow, and hit the road back to New Orleans. As a result, I’m really not up for thinking too hard right now, but I probably won’t even get a chance to post tomorrow, so I’ll take a cute from a commenter on my Intermediate Value Theorem post and mention the “ham sandwich” theorem.
The MathWorld entry is pretty good for references here. Basically, if we have three solids (whatever those are) floating out anywhere in three-dimensional space, like two slices of bread and a slice of ham, then you can cut them all in half with one plane. Or if you have four four-dimensional solids you can cut them all in half by a three-dimensional hyperplane. And so on.
The sketch of the proof on that page is pretty clear, I think. You pick a direction and consider planes perpendicular to it. The IVT gives you one plane for each of the three solids. Then you use these to construct a map from the sphere to the plane whose image you can show must contain the point , which corresponds to all three planes lining up in that direction.
Interestingly, the last step also boils down to something like saying that the image of a connected space under a continuous map is connected. But it’s not quite that. We’ll deal with it later, rest assured.