@MoisheKohan: The induced metric.

@IgorBelegradek: Yes, I changed the word "characterized" to "identified".

@IgorBelegradek: The question is whether $X$ is convex assuming that $\Pi\cap X$ is a convex subset of $\Pi$, for all planes $\Pi$ passing through some point $p$ of $M$. A subset of $\Pi$ (which itself is a Cartan-Hadamard space) is convex if every pair of points of $\Pi$ can be joined by a geodesic of $\Pi$ which lie in that set.

@IgorBelegradek: Thanks, I edited the question to clarify this point.