ex.tex

%%!TEX root = all.tex
\chapter{Exercises}

\begin{thm}{Exercise}
Give an example of metric on a finite set, which satisfies the comparison inequality 
\[\angk{0}{p}{x_1}{x_2}+\angk{0}{p}{x_2}{x_3}+\angk{0}{p}{x_3}{x_1}
\le
2\cdot\pi\]
for any quadruple of points $(p,x_1,x_2,x_3)$, 
but which is not isometric to a subset of Alexandrov space with curvature $\ge0$.
\end{thm}

\begin{description}
\item[Hint:] Consider metric on 5-point set $p,x_1,x_2,x_3,x_4$ such that $|x_ix_j|=a$ and $|p x_i|=b$ for $a$ and $b$ such that $\tangle\mc0(p,x_ix_j)=2\cdot\pi/3$.
\end{description}


\begin{thm}{Flat triangle is not flat} 
\label{ex:not-flat}
Construct a space $\spc{L}\in\CBB{3}0$ with a triangle 
$\trig x y z$ in $\spc{L}$ with all angles equal to the corresponding angles of its model triangle, but which can not be filled by an isometric copy of the model triangle. (Compare to \ref{ex:flat-in-CBB}).
\end{thm}

\begin{thm}{Exercise}
Give an example of Alexandrov space, which has no concave function.
Namely, there is no open subset $\Omega$ which admits a concave function $f\:\Omega\to \RR$.
\end{thm}

\begin{thm}{Exercise}Construct a non-compact space $\spc{L}\in\CBB{}0$ which contains no rays.
\end{thm}