In geometry, a subset of a Euclidean space, or more generally an [[affine-space|affine space]] over all the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them.
Equivalently, a convex set (region) is a subset that intersects every line into a single line segment (possibly empty). For example, a cube is convex but anything that is hollow or has an indent, for example a crescent shape, is not convex.
Figure: Convex shape (left) and non-convex shape (right).
The boundary of a convex set is always a convex curve. The intersection of all convex sets that contain a given subset \(A\) of Euclidean space is called the convex hull of \(A\). It is the smallest convex set containing \(A\) (related to generated \(\sigma\)-algebra).
0.1 Convex Set
Let \(S\) be a vector space or an [[affine-space|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 id, for all \(x\) and \(y\) in \(X\), the line segment connecting \(x\) and \(y\) is included in \(C\).
This means that the [[affine-combination|affine combination]] \[ (1-t)x+ty\in C \] for all \(x,y\in C\) and \(t\in[0,1]\).
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|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|extreme point]].
A set \(C\) is absolutely convex if it is convex and [[balanced-set|balanced]].