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
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