Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space. Lines and segments. Given two points x; y 2 X, we define ! xy = fx + t(y − x): t 2 Rg = f(1 − t)x + ty : t 2 Rg ; [x; y] = fx + t(y − x): t 2 [0; 1]g = f(1 − t)x + ty : t 2 [0; 1]g : The sets xy! and [x; y] are, respectively, the line passing through x and y, and the (closed) segment with endpoints x; y. Observe that they reduce to a singleton whenever x = y. We shall use also the following notation for non-closed segments: (x; y] = [x; y] n fxg; [x; y) = [x; y] n fyg; (x; y) = [x; y] n fx; yg: Linear, affine, and convex sets. A set A ⊂ X is called: • linear if A is a vector subpace of X (i.e., A is nonempty, and αx + βy 2 A whenever x; y 2 A; α; β 2 R); • affine if the line passing through any two points of A is entirely contained in A (i.e., (1 − t)x + ty 2 A whenever x; y 2 A; t 2 R); • convex if any segment with endpoints in A is contained in A (i.e., (1 − t)x + ty 2 A whenever x; y 2 A; t 2 [0; 1]). Obviously, each linear set is affine, and each affine set is convex. Moreover, any translate of an affine (convex, respectively) set is affine (convex, resp.). 2 Example 0.1. A linear set in R is either the singleton f0g, or a line containing 0, 2 or the whole R . 2 2 An affine set in R is either ;, or a singleton, or a line, or R . 2 There is a large variety of convex sets in R ... Proposition 0.2. Let A be a set in a vector space X. (a) A is linear if and only if A is affine and contains 0. (b) A is affine if and only if either A = ; or A is a translate of a linear set. (Moreover, the linear set is unique in this case.) Proof. Exercise. By Proposition 0.2(b), the following definition is justified. The dimension and the codimension of an affine set A ⊂ X is defined as, respectively, the dimension and the codimension of the (unique) linear translate of A. 1 2 Hyperplanes. A set H ⊂ X is a hyperplane in X if it is an affine set of codimension 1. Equivalently, a hyperplane is any maximal proper affine subset of X. Proposition 0.3. A set H ⊂ X is a hyperplane if and only if it is of the form −1 H = ' (α) where ': X ! R is a nonzero linear functional and α 2 R. Proof. \)" Fix any x0 2 H. By Proposition 0.2, the set Y = H − x0 is a linear subspace of codimension 1 in X. Fix any v0 2 X n Y . Then each x 2 X has a unique representation of the form x = yx + txv0 where yx 2 Y , tx 2 R. The mapping '(x) := tx is linear and satisfies x 2 H , x − x0 2 Y , '(x − x0) = 0 , '(x) = '(x0): Thus we can put α = '(x0). \(" It is easy to see that '−1(α) is a translate of the kernel '−1(0). We have to −1 −1 show that ' (0) has codimension 1. Fix some v0 2 X n ' (0) (this is possible since ' 6≡ 0). By substituting v0 by its appropriate multiple, we can suppose that '(v0) = 1. Then every x 2 X can be written in the form −1 x = [x − '(x)v0] + '(x)v0 2 ' (0) + Rv0 since '(x − '(x)v0) = 0. Corollary 0.4. Let H be a hyperplane in X. Then X can be written as a disjoint union X = H [ H+ [ H− in such a way that if x 2 H+ and y 2 H− then [x; y] \ H is a singleton. (The sets H+;H− are the algebraically open halfspaces generated by H.) Proof. Exercise. Hint: take '; α as in Proposition 0.3 and consider H+ = fx 2 X : − '(x) > αg, H = fx 2 X : '(x) < αg. Convex and affine combinations. Definition 0.5. Let A ⊂ X. An affine combination of elements of A is any finite sum of the form n X Pn (1) λixi where xi 2 A, λi 2 R, 1 λj = 1. i=1 A convex combination of elements of A is any finite sum of the form (1) with λi ≥ 0 (i = 1; : : : ; n). Proposition 0.6. Every convex/affine/linear set in a vector space X is closed under making convex/affine/linear combinations of its elements. Proof. The \linear part" is well known from Linear Algebra. Pn Let C be a convex set and x = i=1 λixi be a convex combination of elements of C. We want to prove that x 2 C. Let us proceed by induction with respect to n. For n = 1, we have x = x1 2 C. Now, suppose that the case n = k holds, Pk+1 and consider the case n = k + 1, that is, x = i=1 λixi with xi 2 C, λi ≥ 0, 3 Pk+1 1 λj = 1. If λk+1 = 1 then necessarily x = xk+1 2 C. Suppose λk+1 6= 1. Then Pk s := 1 λj = 1 − λk+1 6= 0. We can write " k # X λi x = (1 − λ ) x + λ x : k+1 s i k+1 k+1 i=1 Since the sum in the quare brackets belongs to C by our induction assumption, x belongs to C. The “affine part" can be proved in the same way. The only difference is that we start with indexing the points xi in such a way that λk+1 6= 1 (if this is not possible, we are in a trivial case). It is easy to see that the set of all convex combinations of elements of fx; yg is the segment [x; y], and the set of all affine combinations of elements of fx; yg is the line xy!. Observation 0.7. A convex/affine combination of convex/affine combinations of elements of A is a convex/affine combination of elements of A. Fact 0.8. Let X be a normed linear space, x; y; z 2 X, z 2 (x; y). Then kz−yk kx−zk kx − yk = kx − zk + kz − yk and z = kx−yk x + kx−yk y : Proof. We have z = (1 − t)x + ty for some t 2 (0; 1). Then z − x = t(y − x) and kz−xk kz−yk z − y = (1 − t)(x − y). Passing to norms we get t = ky−xk and 1 − t = kx−yk . Now the two formulas easily follow. Hulls. It is an easy but important observation that the intersection of any family of linear/affine/convex sets is again a linear/affine/convex set. (The same does not hold for unions, but does holds for linearly ordered unions.) Given a set A ⊂ X, the intersection of all linear sets containing A is the linear hull of A, denoted by span(A). Analogously, we can define the affine hull of A as the intersection of all affine sets containing A, and the convex hull of A as the intersection of all convex sets containing A. The affine and the convex hull of A will be denoted by aff(A) and conv(A) ; respectively. Obviously, A is linear if and only if span(A) = A; A is affine if and only if aff(A) = A; A is convex if and only if conv(A) = A. It is a well known fact that the linear hull of a set A coincides with the set of all linear combinations of elements of A. The following theorem states that analogous properties hold for convex hulls and for affine hulls as well. Theorem 0.9. Let A be a set in a vector space X. Then conv(A) = fx 2 X : x is a convex combination of elements of Ag ; aff(A) = fx 2 X : x is an affine combination of elements of Ag : 4 Proof. Let us prove the first formula (the second one is analogous). By Obser- vation 0.7, the set C of all convex combinations of points of A is convex; thus conv(A) ⊂ C. On the other hand, any point x 2 C, being a convex combination of points of conv(A), belongs to conv(A) by Proposition 0.6. Let A be an affine set in a vector space X of a finite dimension d and let x 2 aff(A). By translation, we can suppose that 0 2 A. In this case, x belongs to the linear hull of A, and hence it is a linear combination of at most d elements of A: d X x = λixi where λi 2 R; xi 2 A: i=1 Since 0 2 A, we can write x as an affine combination of d + 1 points of A: d X Pn x = λ0 · 0 + λixi with λ0 = 1 − 1 λj. i=1 Thus we have proved that, in an d-dimensional vector space, every point of the affine hull of a set is an affine combination of d + 1 or fewer points of A. The following important theorem shows that a similar result holds for convex hulls as well. Theorem 0.10 (Carath´eodory). Let A be a subset of a d-dimensional vector space X. Then conv(A) = fx 2 X : x is a convex combination of d + 1 or fewer points of Ag : Proof.