  This article extends to the vector setting the results of our previous work
Kruger et al. (2015) which refined and slightly strengthened the metric space
version of the Borwein-Preiss variational principle due to Li and Shi, J. Math.
Anal. Appl. 246(1), 308-319 (2000). We introduce and characterize two seemingly
new natural concepts of epsilon-minimality, one of them dependent on the chosen
element in the ordering cone and the fixed "gauge-type" function.
