  An r-unform n-vertex hypergraph H is said to have the
Manickam-Mikl\'os-Singhi (MMS) property if for every assignment of weights to
its vertices with nonnegative sum, the number of edges whose total weight is
nonnegative is at least the minimum degree of H. In this paper we show that for
n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS
property, and the bound on n is essentially tight up to a constant factor. This
result has two immediate corollaries. First it shows that every set of n>10k^3
real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative
k-sums, verifying the Manickam-Mikl\'os-Singhi conjecture for this range. More
importantly, it implies the vector space Manickam-Mikl\'os-Singhi conjecture
which states that for n >= 4k and any weighting on the 1-dimensional subspaces
of F_q^n with nonnegative sum, the number of nonnegative k-dimensional
subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional
generalizations, which can be regarded as analogues of the Erd\H{o}s-Ko-Rado
theorem on k-intersecting families.
