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What is the largest area/volume of any $(d-1)$-dimensional flat object that fits into the $d$-dimensional unit hypercube? For instance, for $d=2$, the answer is $\sqrt 2$, as this is the length of the (1-dimensional) diagonal. But how does this quantity grow as a function of $d$?

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The maximal area of a $d-1$ dimensional slice through a $d$-dimensional hypercube is $\sqrt 2$ in any dimension, this was proven by K.M. Ball, Cube slicing in $\mathbb{R}^n$ (1986).

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