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the union of two convex sets need not be a convex set.
no
yes
The answer depends on how it is halved. If the plane is divided in two by a step graph (a zig-zag line) then it will not be a convex set.
A convex polygon is one with no reflex angles (angles that measure more than 180 degrees when viewed from inside the polygon).
More generally a convex set is on where a straight line between any two points in the set lies completely within the set.