●A>0今-A<0. e If A is a tall full-rank matrix. then A'A>0 and AA>0 ·ⅡfA>0and|Bl≠0, then BAB>0.■ 2. Concavity Given points RE R, a convex combination is y=ArzI 入k≥0 Given any two points a, y, the intervals are ,列≡{2|2=Ax+(1-y,A∈p,1} (x,y)≡{=|2=Mx+(1-My,A∈(0,1)} a set SCRn is a convex set if ,y∈ x,引cS. Proposition 2.2.(Properties of Convex Sets 1. Any intersection of convex sets is also convex 2. The Cartesian product of convex sets is also convex. f: X-R is concave if XCRn is convex and fx+(1-)≥Af(x)+(1-入)f(y),A∈(0,1),x,y∈X If the inequality holds strictly, f is strictly concave f is(strictly) convex if -f is(strictly)concave Theorem 2.2.(Properties of Concave Functions). X CRn is convex 1.f:X→ R is concave iff{(x,t)∈X×R|f(x)≥t} Is convex 2. Concave functions are continuous in the interior of their domains 3. A function f: X-R is concave iff f(A1x2+…+Akx2)≥Af(x2)+…+Akf(x), for all k>1 and all convex combinations A1 +.+Ak.x• A > 0 ⇔ −A < 0. • If A is a tall full-rank matrix, then A0 A > 0 and AA0 ≥ 0. • If A > 0 and |B| 9= 0, then B0 AB > 0.  2. Concavity Given points xk ∈ Rn, a convex combination is y ≡ λ1x1 + ··· + λmxm, λk ≥ 0, [m k=1 λk = 1. Given any two points x, y, the intervals are [x, y] ≡  z | z = λx + (1 − λ)y, λ ∈ [0, 1] , (x, y) ≡  z | z = λx + (1 − λ)y, λ ∈ (0, 1) . A set S ⊂ Rn is a convex set if ∀ x, y ∈ S, [x, y] ⊂ S. Proposition 2.2. (Properties of Convex Sets). 1. Any intersection of convex sets is also convex. 2. The Cartesian product of convex sets is also convex.  f : X → R is concave if X ⊂ Rn is convex and f[λx + (1 − λ)y] ≥ λf(x) + (1 − λ)f(y), ∀ λ ∈ (0, 1), x, y ∈ X. If the inequality holds strictly, f is strictly concave. f is (strictly) convex if −f is (strictly) concave. Theorem 2.2. (Properties of Concave Functions). X ⊂ Rn is convex. 1. f : X → R is concave iff {(x, t) ∈ X × R | f(x) ≥ t} is convex. 2. Concave functions are continuous in the interior of their domains. 3. A function f : X → R is concave iff f(λ1x1 + ··· + λkxk) ≥ λ1f(x1 ) + ··· + λkf(xk), for all k ≥ 1 and all convex combinations λ1x1 + ··· + λkxk.  2—3