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\begin{document}
\begin{minipage}[t]{0.5\textwidth}
\begin{flushleft}
\includegraphics[scale=0.09]{bms-logo-eff}\\[5pt]
{\small Prof.~G\"unter M.~Ziegler} \\
{\small Albert Haase, Miriam Schlöter}
\end{flushleft}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
\begin{flushright}
\includegraphics[scale=0.4]{fu-logo}\\[5pt]
{\small Institut f"ur Mathematik }\\
{\small Arbeitsgruppe Diskrete Geometrie}\\
\end{flushright}
\end{minipage}

\enlargethispage{5mm}
\vspace{20pt}
\centerline{\Large \bf{Discrete Geometry II -- Problem Sheet 11}}
\vspace{15pt}

Please hand in your solutions to Prof.~Ziegler on \textbf{Tuesday, July~8, 2014} before the lecture begins. 

\vspace{30pt}

{\large\bf Problem 1: }\textit{An Order Polytope}\hfill (8 Points)\\[3pt]
Let $n\ge2$, and let $(X,\preceq)$ be a disjoint union of an $(n-1)$-chain and an element.
For example, let $X=\{a_1,a_2,\dots,a_n$ with $a_i\prec a_j$ iff $1<i<j$\}.
The \Emph{order polytope} of $(X,\preceq)$ is
\[
P(X,\preceq) = \{ x \in [0,1]^n : x_i\le x_j\textrm{ for }a_i\preceq a_j\}.
\]
\begin{compactenum}[\normalfont (a)]
\item Describe $P(X,\preceq)$: What is its dimension, its number of facets, its number of vertices, its volume?
\item How many pairwise comparisons do you need for sorting the poset $(X,\preceq)$?\\
    What upper bound for the number of steps do you get from the approach/proof discussed in class?
\end{compactenum}

\vspace{20pt}

{\large\bf Problem 2: }\textit{Type Cones and Minkowski Addition}\hfill (6(+2) Points)\\[3pt]
Let $A\in\R^{n\times d}$ be a rank~$d$ matrix for which the sum of the rows is zero.
\begin{compactenum}[\normalfont (a)]
  \item Show that for any $b\in\R^n$, the polyhedron $P_A(b)$ is bounded.
    \\(More generally, this holds if and only if there are positive values $\alpha_i$
       such that $\alpha_1 a_1+\ldots+\alpha_n a_n=0$.)
  \item Give an example where $P_A(b')+P_A(b'')\neq P_A(b'+b'')$.\\
      If there is no equality, which inclusion fails?
  \item \emph{Bonus:} Show that $P_A(b')+P_A(b'')=P_A(b'+b'')$ if and only if 
       $b'$ and $b''$ lie in the closure of the same type cone (namely, the type cone $\mathcal{T}_A(b'+b'')$).
\end{compactenum}

\newpage
% 

{\large\bf Problem 3: }\textit{An Inner Type Cone}\hfill (8 Points)\\[3pt]
\begin{compactenum}[\normalfont (a)]
 \item   For the matrix
\[A :=\left(\begin{array}{rr}
     1 &  0 \\[-2pt]
     0 &  1 \\[-2pt]
     1 &  1 \\[-2pt]
    -1 &  0 \\[-2pt]
     0 & -1
\end{array}\right)\]
describe 
\begin{compactenum}[\normalfont (i)]
    \item the cone $\B_A$ of all right-hand side vectors $b\in\R^5$ such that $P_A(b)\neq\emptyset$,
          \\Is this a polyhedral cone? What is its dimension? How many facets/rays does it have?
    \item the closed inner region $\B_A^\circ$ 
          of all right-hand side vectors $b\in\R^5$ such that all five inequalities define non-empty faces of $P_A(b)$,
          \\What is its dimension? Is this a polyhedral cone? If yes, how many facets/rays does it have?
    \item the other type cones that $\B_A$ decomposes into? Sketch the decomposition.
\end{compactenum}

\item Argue that for a general matrix $A$, the inner type cone is a polyhedral cone. Give an 
  upper bound for its number of facets.\\
  (Hint: When does an inequality become redundant? If it does, due to “too large right-hand side”,
  then some version of the Farkas lemma will give you a certificate, which you can use.)
\end{compactenum}


\end{document}

