It is known that if a regular N-gon located inside a circle extends all sides to the intersection with this circle, then 2N segments added to the sides can be divided into two groups with the same sum of lengths. Is the same statement true for the one inside the sphere
a) an arbitrary cube?
It is not very clear what you asking, but the counterpart of the two-dimensional result holds for each convex polyhedron $P$ contained inside a sphere $S$, whose faces are equilateral polygons, because for each face $F$, new segments adjacent to sides of $F$ can be split into two parts of equal total length, that follows from the two-dimensional result for the face $F$ and an intersection circle of $S$ with a plane containing $F$.