I guess that for arbitrary $n\in\mathbb{N}$ it is impossible to decompose the open unit ball $B(0,1)$ of $R^n$ in four disjoint sets $A$,$B$,$C$,$D$, such that $A$,$B$,$C$ be convex open subsets and the set $D$ the common boundary for the three, i.e $D=\partial A=\partial B=\partial C$.
I think it is true, but I can not catch the key idea to prove this.
If we do not care about convexity, it is known that it is possible to decompose an open ball of $R^2$ in three disjoint connected parts with common boundary (i.e.Wada Lakes), similar is possible in higher dimensions.
If my guess is not true please provide me with reference or example, If someone see how to prove the claim please provide me with proof or a clue at least.
The boundary of the unit ball must be part of the boundary of the pieces. Hence, if all pieces have the same boundary and are convex, then they must all contain the interior of the unit ball.
If you discard the boundary of the unit ball itself, there is actually a solution. You can partition the unit ball in the points with positive, zero and negative first coordinate. All three parts will have the points with zero first coordinate as boundary.