Homework 3

Due: Oct. 22, 14:00:00

Midpoint convexity

  1. A set CRn is midpoint convex, if for all u,vC, the midpoint (u+v)/2 is also in C. 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.

Separating and supporting hyperplanes

  1. Express the convex set {(x,y)R02:xy1} as an intersection of halfspaces.

Convex functions

  1. The α-sublevel set of a function f:DRnR is defined as Cα={xD:f(x)α}. Show that sublevel sets of a convex function are convex for any value of α.

    In particular, show that the set of global minima of a convex function is convex. Namely, the set M={x:f(x)f(x),xD} is also convex if f is convex.

  2. Suppose f:RnR is convex and bounded above over Rn. Show that f is constant.

  3. Suppose f,g:RnR are two functions defined on Rn, f is convex and g is concave. Show that if g(x)f(x) for all xRn, there exists an affine function h such that g(x)h(x)f(x) for all xRn.