Some simple problems
- Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
- Using the second order condition show that the following functions are convex on the specified domain.
- \( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).
- \( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).
- Let \( f(x) = x^2 \) and \( g(x) = e^x \). Show that \( f(g(x)) \) and \( g(f(x)) \) is convex for \( x \in \mathbb{R} \). Also show that if \( f(x) \) is any convex function than \( h(x) = e^{f(x)} \) is convex.
- A norm is any function that satisfy the following properties
- \( f(\alpha x) = |\alpha| f(x) \) for all \( \alpha \in \mathbb{R} \).
- \( f(x+y) \leq f(x) + f(y) \)
- \( f(x) \leq 0 \) for all \( x \in \mathbb{R}^n \) with equality if and only if \( x = 0 \)
Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).