Seminar 9

11-01, the rest
11-04. (It might be a bit more quirky than 11-02, so you might want to look at that too.)

(As suggested in the draft. Exam set: https://www.uio.no/studier/emner/sv/oekonomi/ECON4140/oldexams/M3x2017.pdf Links to an external site. . Solution: put "SOLVED" before .pdf. Beware that my handwritten solution is a bit terse - as stated in the typeset part, I would recommend you to spend more ink.)

Show that the intersection of an arbitrary family of convex sets, is convex.
- Warning: it may be an arbitrarily large family, in particular larger than what you can put in a sequence, so an induction argument is useless!
- Notation: if J is an arbitrary index set, and for each j in J we have a (convex set), then we can write the family as ${S_{j}}_{j \in J}$ intersection set as $LaTeX: \bigcap_{j\in J}S_j$ $\bigcap_{j\in J}S_j$
Use the epigraph definition to show that the maximum of two convex functions (on the same convex domain D) is a convex function.
- Trick question for the curious: Let $LaTeX: \{f_j\}_{j\in J}$ $\{f_j\}_{j\in J}$ be a family of convex function on the same convex domain D. Does it follow that the smallest f that is $LaTeX: \geq f_j$ $\geq f_j$ for all j, is a convex function?
The convex hull of a set A is "the smallest convex set that is $\supseteq A$ ". That means, the intersection of all sets S that have the following two properties: S is convex and $S \supseteq A$ .
- The phrase "convex hull" surely suggests that this is a convex set - it would be quite bad to give it that name if not. Show that the convex hull is indeed convex.
Let f(t) = t + sin t.
- Is f quasiconvex? Is f quasiconcave?
- For what real constants c will f(t)+ct be quasiconcave? For what c will f(t)+ct be quasiconvex?