# Maximal area/volume of (d-1)-dimensional object in d-dimensional hypercube

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 $$d$$?

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).