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. Such estimates on these
sums in the literature were known only for convex polygons. We also sharpen a
previous lower bound on the minimum sum of pairwise squared distances for
convex polygons due to Novotn\'y.

In the second part, we determine an exact formula for the maximum perimeter
of a simple $n$-gon ($n$ odd) contained in a disk of unit radius. This answers
an open problem posed by Bra\ss. 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.