  Let $A_1,A_2,...,A_n$ be the vertices of a polygon with unit perimeter, that
is $\sum_{i=1}^n |A_i A_{i+1}|=1$. We derive various tight estimates on the
minimum and maximum values of the sum of pairwise distances, and respectively
sum of pairwise squared distances among its vertices. In most cases such
estimates on these sums in the literature were known only for convex polygons.
  In the second part, we turn to a problem of Bra\ss\ regarding the maximum
perimeter of a simple $n$-gon ($n$ odd) contained in a disk of unit radius. The
problem was solved by Audet et al. \cite{AHM09b}, who gave an exact formula.
Here we present an alternative simpler proof of this formula. We then examine
what happens if the simplicity condition is dropped, and obtain an exact
formula for the maximum perimeter in this case as well.
