On equipartitions of surfaces of 3D convex regions

Question

Let S be the surface of a 3D convex region (a 'convex surface'). Let S' be a subset of S. We shall refer to S' as geodetically convex in S if the following condition holds: If A and B are two points in S', (at least one of ) the shortest path(s) on S between A and B contains no point that is not in S'.

Question: It is easy to see that if S is the surface of a sphere and n is any integer, S can be partitioned into n equal area pieces that are also geodetically convex in S - these pieces are separated by great circle arcs between any 2 antipodal points on S. Prolate and oblate spheroids (and perhaps all surfaces got by rotating a curve about an axis) too seem to allow such equipartitions. Are there any other convex surfaces for which such an equipartition exists for any n? If so, how does one characterize such surfaces?

Note 1: If S is the surface of a cube, I suspect it cannot be cut into n geodetically convex equal area pieces for all n. One could also ask which convex surfaces are worst for such equipartitions - ie if there are convex surfaces which do not allow such an equipartition for any n.

Note 2: Even if we drop the equal area requirement and need the n pieces only to be geodetically convex in S, the question appears tricky for general n. For n=2, since a closed convex surface always has a closed geodesic (as per a theorem of Poincare), such a geodesic cuts a convex surface S into 2 pieces that are geodetically convex in S.

Answer 1

Let $C$ be a cone with lateral side $A$ and base $B$. So $C = A \cup B$. The base is geodetically convex. But if I'm interpreting the definition correctly, $A$ is not geodetically convex: For any two points $a$ and $b$ on the rim, the unique shortest path from $a$ to $b$ lies in $B$ (blue in the figure). The geodesic segment on $A$ (red) is always longer.

Perhaps this is the behavior you want. But if not, you might change the definition to:

A set $S$ is convex if, for every pair of points in $S$, there exists a geodesic segment $\gamma$ connecting them which remains entirely within $S$.

I.e., drop the "shortest path(s)" condition.

Even with this definition, there are what might be considered anomalies. Consider the cone $C$ minus the apex $x$: $C \setminus \{x\}$. Then I believe this surface with a point-hole is convex, because no geodesic passes through $x$. Again, maybe this is fine, e.g., for smooth surfaces.