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This is precisely the directional derivative (+) f (x + tu) − f (x) 0 t f (x; u) := lim t of f in direction u. 2. 3. Let f : Rn → (−∞, ∞] be convex and x ∈ int dom f . Then, for each u ∈ Rn , u = 0, the directional derivative f (x; u) of f exists. The corollary does not imply that f (x; u) = −f (x; −u) holds (in fact, the latter − + equation is only true if g(u) (0) = g(u) (0)). , fn (x) of f need not exist in each point x. , fn (x) exist, the function f is even differentiable. Even more, a convex function f on Rn is twice differentiable almost everywhere (in a suitable sense).

N}. A doubly stochastic matrix with components in {0, 1} is called permutation matrix. 5. EXTREMAL REPRESENTATIONS 37 2 (a) The set K ⊂ Rn of doubly stochastic matrices is compact and convex. (b) The extreme points of K are precisely the permutation matrices. Hint for (b): You may use the following simple combinatorial result (marriage theorem): Given a finite set H, a nonempty set D and a function f : H → P(D) with ˜ f (h) ≥ |H|, ˜ ⊂ H, for all H ˜ h∈H then there exists an injective function g : H → D with g(h) ∈ f (h), for all h ∈ H.

4 shows that z ∈ int P , a contradiction. 32 CHAPTER 1. CONVEX SETS Exercises and problems 1. Let A ⊂ Rn be closed and int A = ∅. Show that A is convex, if and only if every boundary point of A is a support point. ∗ 2. Let A ⊂ Rn be closed. Suppose that for each x ∈ Rn the metric projection p(A, x) onto A is uniquely determined. Show that A is convex (Motzkin’s theorem). 3. Let A ⊂ Rn be non-empty, closed and convex. e. with outer normal u). 4. Let A, B ⊂ Rn be non-empty and convex, and assume rel int A ∩ rel int B = ∅.

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