\section{List of Appendices}\label{appendix:table-of-contents}
In the appendices, we first provide a table summarizing the notations defined in the main paper. We then provide the proofs of our theoretical results in full detail in the subsequent sections. 

The remainder of the appendices are summarized as follows: 
\begin{itemize}
\item \appref{appendix: table of notations} provides a list of notations defined in the main paper
\item \appref{appendix: regions relaxed general position} provides the proof of \thmref{thm:d'-general} (Number of  Regions Induced by Intersections of  Halfspaces)
\item \appref{appendix: faces teaching} provides the proof of \propref{faces_teach} (Number of Faces Induced by Intersections of Halfspaces)
\item \appref{appendix: main theorem} provides the proof of \thmref{theorem: Main Theorem} (Teaching Complexity of Convex Polytopes)
\item \appref{appendix: active learning} provides the proof of \thmref{pairwise teaching bound} (Learning Complexity of Convex Polytopes)
\item \appref{appendix: Dual map} provides the proof of \thmref{dual map} and \corref{cor: teach phi surfaces} (Teaching Complexity of $\phi$-Separable Dichotomy)
\item \appref{appendix: extreme points}  provides the proof of \thmref{main extreme points} (Equivalence of Teaching Set and Extreme Points)
\item \appref{appendix: teaching ranking} provides an additional use-case of the problem of teaching convex polytopes via halfspace queries. In particular, we introduce the problem of teaching linear rankings via halfspaces queries, and establish a $\bigTheta{d}$ bound on the average teaching complexity.


\end{itemize}
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