Convex Optimization 2026, 02 - Affine and Convex Sets
Two points in ℝn x1 ≠ x2, and y = θx1 + (1 - θ)x2, θ ∈ ℝ so that it forms the line passing through both points. Whichever point corresponds to y = 0 is the base point, and the line is pointed toward the other (2.1.1). An affine set is a collection of points, where the line between any two of its distinct points is also part of that set (2.1.2). A linear combination of elements of an affine set is the affine combination of its points. They are also part of affine set that its points are part of. The subspace of an affine space that fixes the base point is closed under sums and scalar multiplication. The full affine space can be written as such a subspace, offset by the base point. A set of all affine combinations of points in a subset of ℝn is the affine hull, written as aff C. If S is any affine set with C ⊂ S, aff C ⊂ S (p. 37 - 38). Define the relative interior of the affine set C as relint C = {x ∈ C | B(x, r) ∩ aff C ⊂ C ∀ r > 0}. Complementarily, the relative boundary is cl C \ relint C, where cl C is the closure of C (2.1.3).
A convex set is a set where line segments between any of its two points are inside it. Line segments are limited by its end-points, whereas lines are infinite. Affine sets are thus convex (2.1.4). The convex hull is written conv C and is the set of all convex combinations in C.A set C is a cone (or nonnegative homogeneous) if ∀ x ∈ C, θ ≥ 0, then θx ∈ C. Convex cones just that, so that θ1x1 + θ2x ∈ C. The conic hull of C is the set of all conic combinations of points in C (2.1.5).