A remark on the problem of nonnegative $k$-subset sums

Given a set of $n$ real numbers with a nonnegative sum, consider the family of all its $k$-element subsets with nonnegative sums. How small can the size of this family be? We show that this problem is closely related to a problem raised by Ahlswede and Khachatrian in [1]. The latter, in a special case, is nothing else but the problem of determining a minimal number $c_n(k)$ such that any $k$-uniform hypergraph on $n$ vertices having $c_n(k)+1$ edges has a perfect fractional matching. We show that results obtained in [1] can be applied for the former problem. Moreover, we conjecture that these problems have in general the same solution.