A Set where all the points in the line segment connecting two points in the set, lie in the set.

In more general cases the "line segment" is defined as all Convex combinations of the objects at the end points.

Operations that preserve convexity

Pages 13 and 14 here

• Intersection
• Affine transformation

Separating hyperplane

For every convex set, and any point outside the set (but in some underlying space, in particular $R^n$..), there exists a Hyperplane that separates the set and the point, i.e. the point is one half-space, and the whole set lies on the other. This is called the separating hyperplane.

Pages 16-20 here

Convex hull

If $g$ is a Convex function, then the Set $\{x|g(x) \leq 0 \}$ is convex.