Solution.

The distance from edge $A A_{1}$ to diagonal $B_{1} D$ is equal to the distance from this edge to the plane $B B_{1} D_{1} D$, i.e., the length of segment $A_{1} E$ (Fig. 11.30). Let the edge of the cube be $a$; then from $\triangle A_{1} E D_{1}$ we find $2 d^{2}=a^{2}$, from which $a=d \sqrt{2}$. Therefore, $V=a^{3}=2 d^{3} \sqrt{2}$.

Answer: $2 d^{3} \sqrt{2}$.
11.035. Find the volume of a cube if the distance from its diagonal to a non-intersecting edge is $d$.

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