Convex Polygons

Erdös-Szekeres conjecture ★★★

Author(s): Erdos; Szekeres

\begin{conjecture} Every set of $2^{n-2} + 1$ points in the plane in general position contains a subset of $n$ points which form a convex $n$-gon. \end{conjecture}

Convex 'Fair' Partitions Of Convex Polygons ★★

Author(s): Nandakumar; Ramana

\textbf{Basic Question:} Given any positive integer \emph{n}, can any convex polygon be partitioned into \emph{n} convex pieces so that all pieces have the same area and same perimeter?

\textbf{Definitions:} Define a \emph{Fair Partition} of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a \emph{Convex Fair Partition}.

\textbf{Questions:} 1. (Rephrasing the above 'basic' question) Given any positive integer \emph{n}, can any convex polygon be convex fair partitioned into n pieces?

2. If the answer to the above is \emph{"Not always''}, how does one decide the possibility of such a partition for a given convex polygon and a given \emph{n}? And if fair convex partition is allowed by a specific convex polygon for a give \emph{n}, how does one find the \emph{optimal} convex fair partition that \emph{minimizes} the total length of the cut segments?

3. Finally, what could one say about \emph{higher dimensional analogs} of this question?

\textbf{Conjecture:} The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: \emph{Every} convex polygon allows a convex fair partition into \emph{n} pieces for any \emph{n}

Keywords: Convex Polygons; Partitioning