Given data that is labeled as outliers, how can I classify data as outliers?
If you are sure that your data are actually normally distributed and that your outliers actually form a high-dimensional ring around your "good" data, you simply require a distance-metric, e.g. Mahalanobis distance is suitable for normal data, and determine the threshold value for distance from the mean to consider all points beyond the high-dimensional ball or ellipsoid depending on the variance structure outliers according to your description.