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Motivated by two definitions of distance, “pre-symmetric w-distance” and “w-cone distance”, we define the concept of a pre-symmetric w-cone distance in a TVS-CMS and introduce its properties and examples. Also, we discuss the TVS-cone version of the recent results obtained by Romaguera and Tirado. Meanwhile, using Minkowski functionals, we show the equivalency between some consequences concerning a pre-symmetric w-distance in a usual metric space and a pre-symmetric w-cone distance in a TVS-CMS. Then, some types of various w-cone-contractions and the relations among them are investigated. Finally, as an application, a characterization of the completeness of TVS-cone metric regarding pre-symmetric concept is performed, which differentiates our results from former characterizations.
Karimizad et al. (Wed,) studied this question.