\part{Convex sets}

\chapter{Convex sets}
\begin{definition}
A \emph{convex set}
\[S \text{ is convex } \iff \forall \lambda \in [0,1] (\mathbf{x},\mathbf{y} \in S \implies \lambda \mathbf{x}+(1-\lambda)\mathbf{y} \in S) \]
\end{definition}


\begin{definition}
A \emph{closed half space}
\end{definition}

\chapter{Properties of convex sets}

\begin{definition}
An \emph{extreme point of a convex set}
\end{definition}

\begin{lemma}
Closed half spaces are convex sets 
\end{lemma}

\begin{theorem}
The intersection of a countable or uncountable amount of convex sets is convex.
\end{theorem}








-convex hull
- convex hulls are open
- set minux