  We prove that equally spaced choreography solutions of a large class of
$n$-body problems including the classical $n$-body problem and a subset of
quasi-homogeneous $n$-body problems, have equal masses if the dimension of the
space spanned by the point masses is $n-1$, $n-2$, or, if $n$ is odd, if the
dimension is $n-3$. If $n$ is even and the dimension is $n-3$, then all masses
with an odd label are equal and all masses with an even label are equal.
Additionally, we prove that the same results hold true for any solution of an
$n+1$-body problem for which $n$ of the point masses behave like an equally
spaced choreography and the $n+1$st point mass is fixed at the origin.
Furthermore, we deduce that if the curve along which the point masses of a
choreography move has an axis of symmetry, the masses have to be equal if $n=3$
and that if $n=4$, if three of the point masses behave as stated and the fourth
mass is fixed at a point, the masses of the first three point masses are all
equal. Finally, we prove for the $n$-body problem in spaces of negative
constant Gaussian curvature that if $n<6$, $n\neq 4$, equally spaced
choreography solutions have to have equal masses, and for $n=4$ the even
labeled masses are equal and the odd labeled masses are equal and that the same
holds true for the $n$-body problem in spaces of positive constant Gaussian
curvature, as long as the point masses do not move along a great circle.
Additionally, we show that these last two results are also true for any
solution to the $n+1$-body problem in spaces of negative constant Gaussian
curvature and the $n+1$-body problem in spaces of positive constant Gaussian
curvature respectively, for the case that $n$ of the point masses behave like
an equally spaced choreography and the $n+1$st is fixed at a point.
