%
% trennung.tex -- slide template
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Trennung}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick]

\coordinate (u) at (3.5,4.5);
\coordinate (v) at (2.5,2);
\coordinate (va) at ({(3.5/2.5)*2.5},{(3.5/2.5)*2});

\uncover<3->{
\fill[color=darkgreen!20] (0,0) rectangle (5.3,5.3);
\node[color=darkgreen] at (1.5,4.9) {$u\not\ge w$};
\node[color=darkgreen] at (4.4,0.6) {$u\not\ge w$};
}

\uncover<5->{
\begin{scope}
\clip (0,0) rectangle (5.3,5.3);
\draw[color=darkgreen] (0,0) -- ($3*(v)$);
\end{scope}

\node[color=darkgreen] at ($1.2*(va)$)
	[below,rotate={atan(2/2.5)}] {$(1+\mu)v$};
}

\uncover<2->{
	\fill[color=red!20] (0,0) rectangle (u);
}

\fill[color=red] (u) circle[radius=0.08];
\node[color=red] at (u) [above right] {$u$};

\uncover<4->{
	\fill[color=blue!40,opacity=0.5] (0,0) rectangle (v);
}

\uncover<2->{
	\fill[color=blue] (v) circle[radius=0.08];
	\node[color=blue] at (v) [above] {$v$};
}

\uncover<4->{
	\draw[color=blue] (0,0) -- (va);

	\fill[color=blue] (va) circle[radius=0.08];
	\node[color=blue] at (va) [above left] {$(1+\varepsilon)v$};
}

\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}];
\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}];

\uncover<2->{
	\draw[->,color=red] (3.0,-0.2) -- (3.0,1.5);
	\node[color=red] at (3.0,-0.2) [below]
		{$\{w\in\mathbb{R}^n\;|\; w<u\}$};
}

\end{tikzpicture}
\end{center}
\end{column}
\begin{column}{0.48\textwidth}
\begin{block}{Satz}
$u>v\ge 0$\uncover<4->{, dann gibt es $\varepsilon>0$ mit
\[
u\ge (1+\varepsilon)v
\]}%
\uncover<5->{und für $\mu>\varepsilon$ ist 
\[
u \not\ge (1+\mu)v 
\]}
\uncover<6->{%
\begin{proof}[Beweis]
\begin{itemize}
\item<7->
$u>v$ $\Rightarrow$ $u_i/v_i>1$ falls $v_i>0$
\item<8->
\[
\vartheta = \min_{v_i\ne 0} \frac{u_i}{v_i} > 1
\]
\uncover<9->{$\varepsilon = \vartheta - 1$}
\end{itemize}
\end{proof}}
\end{block}
\end{column}
\end{columns}
\end{frame}
\egroup