A set is

*midpoint convex*, if for all , the midpoint is also in . Obviously convex sets are also midpoint convex.Do you think midpoint convex sets are also convex?

If not, please give a counterexample that is midpoint convex but not convex, and then add some mild conditions (like continuity for midpoint convex functions) to make midpoint convex sets become convex.

Prove your claim.

The -

*sublevel set*of a function is defined as . Show that sublevel sets of a convex function are convex for any value of .In particular, show that the set of global minima of a convex function is convex. Namely, the set is also convex if is convex.

Suppose is convex and bounded above over . Show that is constant.

Suppose are two functions defined on , is convex and is concave. Show that if for all , there exists an affine function such that for all .