Question: Let $Q_n=[-\frac 12,\frac 12]^n\subseteq {\bf R}^n$ be the cube in ${\bf R}^n$ centered at the origin and having $n$-dimensional volume equal to one.  What is the maximum $(n-1)$-dimensional volume $M(d)$ of $(H\cap Q_n)$ when $H \subseteq {\bf R}^n$ is a hyperplane?

Can you guess the behavior of $M(n)$ when $n \to \infty$? Can you guess the plane which maximizes the area of intersection for $n=3$?

Answer:  Keith Ball proved that the maximum volume of the intersection of the cube with a hyperplane  in every dimension $n>1$ is $\sqrt 2$.

(Here is a related paper  by Don Chakerian and Dave Logothetti on slices of cubes.)

5 thoughts on “Test Your Intuition (2)”

1. 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” .