Question: Let be the cube in centered at the origin and having -dimensional volume equal to one. What is the maximum -dimensional volume of when is a hyperplane?
Can you guess the behavior of when ? Can you guess the plane which maximizes the area of intersection for ?
Test your intuition before reading the rest of the entry.
Answer: Keith Ball proved that the maximum volume of the intersection of the cube with a hyperplane in every dimension is .
(Here is a related paper by Don Chakerian and Dave Logothetti on slices of cubes.)