Exercise 6.1.4

Answers

(a)
If $x, y \in K_{1} \cap K_{2}$ , then $x, y \in K_{1}$ and $x, y \in K_{2}$ , so any points between $x, y$ will be in $K_{1}, K_{2}$ as well, so they belong to $K_{1} \cap K_{2}$ , which shows the intersection is also convex
(a)
Since $F_{1}, F_{2}$ are convex function, let the set of points on and above each graph be $S_{1}, S_{2}$ respectively, and we know that $S_{1}, S_{2}$ are convex as well. The intersection of $S_{1} \cap S_{2}$ is the set of points on and above the maximum of $F_{1}, F_{2}$ , i.e. above $F_{3}$ , since the intersection is convex, we see that $F_{3}$ is convex

niuers

2020-03-20 00:00