One-dimensional minimal fillings with negative edge weights

A.O. Ivanov and A.A. Tuzhilin proposed a particular case of Gromov??s minimal fillings problem generalized to the case of stratified manifolds using weighted graphs with a nonnegative weight function as minimal fillings of finite metric spaces. In this paper we introduce generalized minimal fillings, i.e., the minimal fillings where the weight function is not necessarily nonnegative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of its generalized fillings.