A set

is$C\subseteq {\mathbb{R}}^{n}$ *midpoint convex*, if for all , the midpoint$u,v\in C$ is also in$(u+v)/2$ . Obviously convex sets are also midpoint convex.$C$ 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.

- Express the convex set
as an intersection of halfspaces.$\{(x,y)\in {\mathbb{R}}_{\ge 0}^{2}:xy\ge 1\}$

The

-$\alpha $ *sublevel set*of a function is defined as$f:D\subseteq {\mathbb{R}}^{n}\to \mathbb{R}$ . Show that sublevel sets of a convex function are convex for any value of${C}_{\alpha}=\{x\in D:f(x)\le \alpha \}$ .$\alpha $ In particular, show that the set of global minima of a convex function is convex. Namely, the set

is also convex if$M=\{{x}^{\ast}:f({x}^{\ast})\le f(x),\mathrm{\forall}x\in D\}$ is convex.$f$ Suppose

is convex and bounded above over$f:{\mathbb{R}}^{n}\to \mathbb{R}$ . Show that${\mathbb{R}}^{n}$ is constant.$f$ Suppose

are two functions defined on$f,g:{\mathbb{R}}^{n}\to \mathbb{R}$ ,${\mathbb{R}}^{n}$ is convex and$f$ is concave. Show that if$g$ for all$g(x)\le f(x)$ , there exists an affine function$x\in {\mathbb{R}}^{n}$ such that$h$ for all$g(x)\le h(x)\le f(x)$ .$x\in {\mathbb{R}}^{n}$