  Let $\mathcal{W}^{n}$ be the class of $C^{\infty }$ complete simply connected
$n-$dimensional manifolds without conjugate points. The hyperbolic space as
well as Euclidean space are good examples of such manifolds. Let $% W\in
\mathcal{W}^{n}$ and let $A$ be a subset of $W$. This article aims at
characterization and building convex and starshaped sets in this class from
inside. For example, it is proven that, for a compact starshaped set, the
convex kernel is the intersection of stars of extreme points only. Also, if a
closed unbounded convex set $A$ does not contain a totally geodesic
hypersurface and its boundary has no geodesic ray, then $A$ is the convex hull
of its extreme points. This result is a refinement of the well-known
Karein-Millman theorem.
