Question: Let ![Q_n=[-\frac 12,\frac 12]^n\subseteq {\bf R}^n](https://s0.wp.com/latex.php?latex=Q_n%3D%5B-%5Cfrac+12%2C%5Cfrac+12%5D%5En%5Csubseteq+%7B%5Cbf+R%7D%5En&bg=ffffff&fg=333333&s=0 "Q_n=[-\frac 12,\frac 12]^n\subseteq {\bf R}^n")
Can you guess the behavior of 
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 
(Here is a related paper by Don Chakerian and Dave Logothetti on slices of cubes.)
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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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