Outline

basic properties and examples
operations that preserve convexity
the conjugate function
quasiconvex functions
log-concave and log-convex functions
convexity with respect to generalized inequalities

1.Definition

$f:\mathbb{R}^n\to\mathbb{R}$ is convex if $\textbf{dom} f$ is a convex set and
$f(\theta x+(1-\theta)y)\leq\theta f(x)+(1-\theta)f(y)$
for all $x,y\in\textbf{dom} f,0\leq\theta\leq 1$

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