[SOLVED] ECE509 Homework1-Convex functions

1. Consider convex functions f_i : Rⁿ →R, i = 1,·· ,k. Prove that the set

{x | fi(x) ≤ 0}

is convex.

2. Consider a convex function f : Rⁿ →R. Prove that the set

{(x,t) | f(x) ≤ t}

is convex.

3. (a) If M₁,M₂ ∈S² are positive definite, prove that M₁ + M₂ is positive definite.

(b) Prove that the set of all n × n positive definite symmetric matrices is convex.

4. Prove that the following set is convex: !
5. Find a necessary and sufficient condition under which the following quadratic function is convex:

.