Elements of Convex Analysis

Proposition 2.0 (i) Any set K ⊂ B is closed and convex iff IK is closed and convex. (ii) Any function f : B →]−∞,+∞] is lower semicontinuous and convex iff epi(f) is closed and convex. (iii) If {Ki}i∈I is a family of closed convex subsets of B, then ⋂ iKi is closed and convex. (iv) If {fi}i∈I is a family of lower semicontinuous convex functions B →] −∞,+∞], then their upper hull f(·) := supi fi(·) is lower semicontinuous and convex.