  We propose a geometric structure induced by any given convex polygon $P$,
called $Nest(P)$, which is an arrangement of $\Theta(n^2)$ line segments, each
of which is parallel to an edge of $P$, where $n$ denotes the number of edges
of $P$. We then deduce six nontrivial properties of $Nest(P)$ from the
convexity of $P$ and the parallelism of the line segments in $Nest(P)$. Among
others, we show that $Nest(P)$ is a subdivision of the exterior of $P$, and its
inner boundary interleaves the boundary of $P$. They manifest that $Nest(P)$
has a surprisingly good interaction with the boundary of $P$. Furthermore, we
study some computational problems on $Nest(P)$. In particular, we consider
three kinds of location queries on $Nest(P)$ and answer each of them in
(amortized) $O(\log^2n)$ time. Our algorithm for answering these queries avoids
an explicit construction of $Nest(P)$, which would take $\Omega(n^2)$ time.
  By applying the aforementioned six properties altogether, we find that the
geometric optimization problem of finding the maximum area parallelogram(s) in
$P$ can be reduced to answering $O(n)$ aforementioned location queries, and
thus be solved in $O(n\log^2n)$ time. This application will be reported in a
subsequent paper.
