By Weil W.

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By part (a), f ∗∗ is closed, hence f ∗∗ ≤ cl f . 1. PROPERTIES AND OPERATIONS OF CONVEX FUNCTIONS 45 Finally, we mention a canonical possibility to describe convex sets A ⊂ Rn by convex functions. The common way to describe a set A is by the function 1A (x) = 1 0 x ∈ A, x∈ / A, if however, 1A is neither convex nor concave. Therefore, we here define the indicator function δA of a (arbitrary) set A ⊂ Rn by δA (x) = 0 ∞ x ∈ A, x∈ / A. if Remark. A is convex, if and only if δA is convex. Exercises and problems 1.

8. A polytope P is polyhedral. Proof. g. that 0 ∈ E := aff P . Also, it is ˜j. ˜ 1, . . e. m P = Hi , i=1 where Hi ⊂ E are k-dimensional half-spaces, then m r ˜j, H ⊥ (Hi ⊕ E ) ∩ P = i=1 j=1 hence P is polyhedral in Rn . Therefore, it is sufficient to treat the case dim P = n. Let F1 , . . , Fm be the faces of P and H1 , . . e. half-spaces with P ⊂ Hi and Fi = P ∩ bd Hi , i = 1, . . , m). Then we have P ⊂ H1 ∩ · · · ∩ Hm =: P . Assume, there is x ∈ P \ P . We choose y ∈ int P and consider [y, x] ∩ P .

4. Let K ∈ Kn . Then K is a polytope, if and only if hK is piecewise linear. Proof. K is a polytope, if and only if K = conv {x1 , . . , xk }, for some x1 , . . , xk ∈ Rn . ,k which holds, if and only if hK is piecewise linear. ,k i = 1, . . , k. Conversely, if hK is linear on the cone Ai , then xi is determined by xi , · = hK on Ai . 58 CHAPTER 2. CONVEX FUNCTIONS Exercises and problems 1. Let f : Rn → R be positively homogeneous and twice continuously partially differentiable on Rn \{0}. Show that there are nonempty, compact convex sets K, L ⊂ Rn such that f = hK − hL .