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I'm not a mathematician, so please feel free to improve my terminology.

In 2D, I am wondering if it is always possible to bisect two non-intersecting, non-overlapping convex arbitrary shapes with a single line no matter the position or orientation of the shapes.

For example, see the image below. I want the line to cut the blue shape into two equal pieces. Same for the red shape.

It seems pretty obvious to me that this should always be possible, but how would one prove it?

This question came up while I was cutting slices of garlic bread with a knife. I cut two pieces at a time to speed up the process. I have to cut the pieces almost exactly in half so my kids won't argue about who got the most. :-)

enter image description here

James
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  • Yes, it is always possible: https://math.stackexchange.com/questions/1389985/bisection-theorem – Vivianne May 19 '21 at 02:40
  • @Vivianne: The question and answer in your link seem to only be discussing a SINGLE bounded region. Again, I'm not a mathematician, so maybe I'm missing its significance. – James May 19 '21 at 02:44

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You can and this generalizes to $n$-dimensional spaces as well. It's called the ham sandwich theorem and the two dimensional case is sometimes called the pancake theorem. In two dimensions you can use a "rotating knife" argument using the Intermediate Value theorem to ensure the existence of such a cut. Higher dimensional proofs require a bit more machinery but are essentially the same idea.

CyclotomicField
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  • Thanks for the ham sandwich theorem link that exactly answers my question! It's really cool that it works in any number of dimensions. – James May 19 '21 at 11:22