information:extraction
Definitions

Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field (this includes Euclidean spaces, which are affine spaces). A subset C of S is convex if, for all x and y in C, the line segment connecting x and y is included in C.

This means that the affine combination (1 − t) x + ty belongs to C for all x,y in C and t in the interval. This implies that convexity is invariant under affine transformations. Further, it implies that a convex set in a real or complex topological vector space is path-connected (and therefore also connected).

A set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the topological interior of C. A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point.

A set C is absolutely convex if it is convex and balanced.