Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property quickly identifies convex subsets of $\mathbf{R}^n$. Indeed, given $x$, $x'\in X$, let $\Pi$ be a plane passing through $p$, $x$, $x'$. Then the line segment $xx'\subset\Pi\cap X\subset X$.
Can convex subsets be identified similarly in a Cartan-Hadamard manifold $M$, i.e., a complete simply connected Riemannian space with nonpositive curvature? A subset of $M$ is convex if it contains the geodesic segment connecting every pair of its points, and we can define the planes $\Pi$ as the image under the exponential map of two dimensional subspaces of $T_p M$. Then $\Pi$ itself is a Cartan-Hadamard space, and we assume that $\Pi\cap X$ is a convex subset of $\Pi$.
In the case where $M$ is the hyperbolic space $\mathbf{H}^n$, the proof in $\mathbf{R}^n$ works just as well, because planes in $\mathbf{H}^n$ are totally geodesic. In a generic manifold, however, there exists no totally geodesic submanifold of dimension bigger than one.
