\part{Convex functions}

\chapter{Convex functions}

\begin{definition}
A \emph{convex function}
\end{definition}

\begin{definition}
A \emph{concave function}
\end{definition}

\begin{proposition}
$f$ is convex iff $-f$ is concave.
\end{proposition}


\chapter{Properties of convex functions}

We start by observing convex functions on open sets of $\mathbb{R}$, specifically in relation to continuity and differentiability.


A real multivariate function with continuous second derivatives is convex iff all principle minors of its Hesian matrix are nonnegative.

A real multivariate function with continuous second derivatives is concave iff all $k$th principle minors of its Hesian matrix have the same sign as $(-1)^k$.

