MathDB

We say that a real number

V

is good if there exist two closed convex subsets

X

Y

of the unit cube in

\mathbb{R}^3

, with volume

V

each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of

X

and

Y

onto that plane are disjoint.
Find

\sup \{V\mid V\ \text{is good}\}

9