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4). 1 Theorem. Intersection. 1] [232, 2] The intersection of an arbitrary collection of convex sets is convex. 1. CONVEX SET This theorem in converse is implicitly false in so far as a convex set can be formed by the intersection of sets that are not. Vector sum of two convex sets C1 and C2 C1 + C2 = {x + y | x ∈ C1 , y ∈ C2 } (19) is convex. By additive inverse, we can similarly define vector difference of two convex sets C1 − C2 = {x − y | x ∈ C1 , y ∈ C2 } (20) which is convex. 2) the set K − K constitutes its affine hull.

N R+ = {x ∈ Rn | xi ≥ 0 ∀ i} (8) The nonpositive orthant Rn− or Rn×n (analogue to quadrant III) denotes − negative and 0 entries. 3 is easily verified by definition (1). 4 affine set A nonempty affine set (from the word affinity) is any subset of Rn that is a translation of some subspace. 2), subspace, etcetera. 4 subspace M and any point x ∈ A A is affine ⇔ A = x + M = {y | y − x ∈ M} (9) The intersection of an arbitrary collection of affine sets remains affine. 1) is the smallest affine set containing it.

6 empty set versus empty interior Emptiness ∅ of a set is handled differently than interior in the classical literature. , paper in the real world. 5 An ordinary flat sheet of paper is an example of a nonempty convex set in R3 having empty interior but relatively nonempty interior. 1 relative interior We distinguish interior from relative interior throughout. 24] and it is always possible to pass to a smaller ambient Euclidean space where a nonempty set acquires an interior. 3]. Given the intersection of convex set C with an affine set A rel int(C ∩ A) = rel int(C) ∩ A (13) If C has nonempty interior, then rel int C = int C .

### Analytic geometry and principles of algebra by Alexander Ziwet, Louis Allen Hopkins

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