  In 1959, Erd\H{o}s and Moser asked for the maximum number of unit distances
that may be formed among the vertices of a convex $n$-gon; until now, the best
known upper bound has been $2\pi n \log_2 n + O(n)$, achieved by F\"uredi in
1990. In this paper, we examine two properties that any convex polygon must
satisfy and use them to prove several new facts related to the question posed
by Erd\H{o}s and Moser. In particular, we improve on F\"uredi's result, and
instead obtain a bound of $n \log_2 n + O(n)$; we exhibit a class of `cycles'
formed by unit distances that are forbidden in convex polygons; and we provide
a lower bound that shows the limitations of our methods. The second result
addresses a question asked by Fishburn and Reeds regarding the possible
configurations of vertices that form a convex polygon.
