I am thinking about the following questions about the cutlocus of a point in a Riemannian manifold or of a hypersurface in the Euclidean space:

1) If all the points of the (nonvoid) cutlocus of a point $p$ in a Riemannian manifold are also conjugate points, then the manifold is rotationally symmetric around the point $p$ and the cutlocus is a single point?

2) If all the points of the (nonvoid) cutlocus of a smooth, connected hypersurface embedded in ${\mathbb R}^{n+1}$ are also conjugate points, what such a hypersurface looks like?

One guess is that it should be a level set of the distance function from the cutlocus itself. In the case $n=1$, that is, curves in the plane, it is a circle.

It can be seen that in this case, by Sard theorem, the cutlocus has Hausdorff $H^n$-measure zero.