  The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices
are the intersection points of consecutive shortest diagonals of $P$. This map
is known to interact nicely with Poncelet polygons, i.e. polygons which are
simultaneously inscribed in a conic and circumscribed about a conic. A theorem
of R. Schwartz says that if $P$ is a Poncelet polygon, then the image of $P$
under the pentagram map is projectively equivalent to $P$. In the present paper
we show that in the convex case this property characterizes Poncelet polygons:
if a convex polygon is projectively equivalent to its pentagram image, then it
is Poncelet. The proof is based on the theory of commuting difference
operators, as well as on properties of real elliptic curves and theta
functions.
