Linear relation: Ax
Affine relation: Ax+b
Affine Set
- contains the line through any two distinct points in the set:
- can be expressed as the solution set of a system of linear equations
Convex Set
- If two points are in the set, then the line segment as well.
- Affine set with 0<a<1
Convex Combinations
- Consider any number of points, with their coefficient being positive and summing up to 1
Convex Hull
- All convex combinations of any points from the set.
Conic (non-negative) combination
- Like convex combinations but without the constraint of "summing up to 1"
- Formed with any points together with the origin.
Conic Hull
- All conic combinations of any points from the set.
Hyperplane and halfspace
- hyperplanes are affine and convex:
- halfspaces are convex
Polyhedron/polytope
- solution space of a system of inequalities like in a linear program
- intersections of a finit set of hyperplanes and halfspaces
Operations that preserves covexity
- S is convex set -> f(S) too, f^-1(S) as well. with f an affine function
- scale, translation, projection...