**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.)

Of course, Keith Ball’s result is related to a large body of mathematics, result and problems, in convexity theory and other areas. Among other things it is related to the well-known “Busemann-Petty Problem” .

Dear Prof Kalai,

you might be interested in this article by Prof Krishna Athreya titled Unit ball in Higher Dimensions

http://www.ias.ac.in/resonance/April2008/p334-342.pdf

Pingback: Test Your Intuition (4) « Combinatorics and more

Pingback: Answer To Test Your Intuition (4) « Combinatorics and more

Pingback: Test Your Intuition (10): How Does “Random Noise” Look Like. « Combinatorics and more