Let $X$ be a finite set with $n$ elements. A function $f:X\too \RR$ such that $\sum_{x\in X} f(x)\ge0$ is called a $n$--weight function. In 1988 Manickam and Singhi conjectured that, if $d$ is a positive integer and $f$ is a $n$--weight function with $n\ge 4d$ there exist at least $n-1\choose d-1$ subsets $Y$ of $X$ with $|Y|=d$ for which $\sum_{y\in Y} f(y)\ge0$. In this paper we study this conjecture and we show that it is true if $f$ is a $n$--weight function and $\left|\{x\in I_n:f(x)\ge0 \}\right|\le d\le {n\over 2}$.
On a problem concerning the weight functions
CHIASELOTTI, Giampiero
2002-01-01
Abstract
Let $X$ be a finite set with $n$ elements. A function $f:X\too \RR$ such that $\sum_{x\in X} f(x)\ge0$ is called a $n$--weight function. In 1988 Manickam and Singhi conjectured that, if $d$ is a positive integer and $f$ is a $n$--weight function with $n\ge 4d$ there exist at least $n-1\choose d-1$ subsets $Y$ of $X$ with $|Y|=d$ for which $\sum_{y\in Y} f(y)\ge0$. In this paper we study this conjecture and we show that it is true if $f$ is a $n$--weight function and $\left|\{x\in I_n:f(x)\ge0 \}\right|\le d\le {n\over 2}$.File in questo prodotto:
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