Test Your Intuition (2)

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?

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 n>1 is \sqrt 2.

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

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5 Responses to Test Your Intuition (2)

  1. Gil Kalai says:

    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” .

  2. anon says:

    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

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

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

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