Reserve in packing cubes into the unit cube

Any collection of $d$-dimensional cubes of total volume not greater than $2^{1-d}$ can be packed into the unit $d$-dimensional cube $I^d$ so that the uncovered part of $I^d$ contains a cube of edge length $1-\sqrt[d]{2}/2$. To achieve this result, the well-known packing method described by Meir and Moser is used.