A minimum problem for finite sets of real numbers with nonnegative sum.

*(English)*Zbl 1273.11045Summary: Let \(n\) and \(r\) be two integers such that \(0 < r \leq n\); we denote by \(\gamma(n, r)\) \([\eta(n, r)]\) the minimum [maximum] number of the nonnegative partial sums of a sum \(\sum^n_{1=1} a_i \geq 0\), where \(a_1, \dots, a_n\) are \(n\) real numbers arbitrarily chosen in such a way that \(r\) of them are nonnegative and the remaining \(n - r\) are negative. We study the following two problems:

(P1) which are the values of \(\gamma(n, r)\) and \(\eta(n, r)\) for each \(n\) and \(r\), \(0 < r \leq n\)?

(P2) if \(q\) is an integer such that \(\gamma(n, r) \leq q \leq \eta(n, r)\), can we find \(n\) real numbers \(a_1, \dots, a_n\), such that \(r\) of them are nonnegative and the remaining \(n - r\) are negative with \(\sum^n_{1=1} a_i \geq 0\), such that the number of the nonnegative sums formed from these numbers is exactly \(q\)?

(P1) which are the values of \(\gamma(n, r)\) and \(\eta(n, r)\) for each \(n\) and \(r\), \(0 < r \leq n\)?

(P2) if \(q\) is an integer such that \(\gamma(n, r) \leq q \leq \eta(n, r)\), can we find \(n\) real numbers \(a_1, \dots, a_n\), such that \(r\) of them are nonnegative and the remaining \(n - r\) are negative with \(\sum^n_{1=1} a_i \geq 0\), such that the number of the nonnegative sums formed from these numbers is exactly \(q\)?

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\textit{G. Chiaselotti} et al., J. Appl. Math. 2012, Article ID 847958, 15 p. (2012; Zbl 1273.11045)

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