It's possible for points and disks, but not for segments. First a counterexample for segments: let $ABC$ be a triangle, and consider points $D, E, F$ on the rays $(CB,(BA,(AC$ respectively such that $A$ lies between $B$ and $F,\ B$ between $C$ and $D$, and $C$ between $A$ and $E$. Our segments will be $AF, BF, CD$, which we move slightly so that they do not intersect anymore. A picture might be helpful. For points, there is the Voronoi diagram: the convex chamber containing the point $P$ in our collection is defined to be the closure of the set of all the points in the plane which are closer to $P$ than to any other point in the collection. It can also be described as the subset of the plane containing $P$ and bounded by all the perpendicular bisectors of segments of the form $PQ$, as $Q\ne P$ ranges through the points in our collection. For disks, we modify the idea used in the construction of Voronoi diagrams: for each disk $D$ in the collection, define the convex set which contains $D$ to be the subset of the plane which contains $D$ bounded by the radical axes of the pairs $D,E$ as $E\ne D$ ranges through the disks in the collection.