A SEMI-DISTANCE AND PROXIMAL DISTANCE ASSOCIATED WITH SYMMETRIC CONE

Recently, there was a distance function d studied in [16], which was shown a semi-distance in the setting of symmetric cone. This result was indeed verified under the assumption of operator commute. It was further proved that it could become a proximal distance in the setting of second-order cone. In this short paper, we improve these results by showing that the distance function d studied in [16] is not only a semi-distance, but also a proximal distance in the setting of symmetric cone without assuming operator commute.