2018-64.md

course	course_year	question_number	tags	title	year
Optimization	IB	64	IB 2018 Optimization	Paper 2, Section $I$, H	2018

What does it mean to state that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a convex function?

Suppose that $f, g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ are convex functions, and for $b \in \mathbb{R}$ let

$$\phi(b)=\inf {f(x): g(x) \leqslant b}$$

Assuming $\phi(b)$ is finite for all $b \in \mathbb{R}$, prove that the function $\phi$ is convex.