This is actually the sandwich problem, which states there is exactly one slice that will split a sandwich of 3 elements into exactly 2 halves regardless of the shape or position of those elements. We don’t need the full proof, but the problem is continuous, so any desired ratio is possible, therefore you will always be able to slice an apple into exactly 1/3 and 2/3rds “good bits”, so a single slice will always be able to do the job.
I still struggle to visualize it. If we have two concentric spheres (or circles), how can you make a cut that slices both into ratios of 2/3 by volume/area?
This is actually the sandwich problem, which states there is exactly one slice that will split a sandwich of 3 elements into exactly 2 halves regardless of the shape or position of those elements. We don’t need the full proof, but the problem is continuous, so any desired ratio is possible, therefore you will always be able to slice an apple into exactly 1/3 and 2/3rds “good bits”, so a single slice will always be able to do the job.
I still struggle to visualize it. If we have two concentric spheres (or circles), how can you make a cut that slices both into ratios of 2/3 by volume/area?