Principal Points of Symmetric, Strongly Unimodal Distributions
Examples illustrate that splitting a symmetric, unimodal distribution into two groups so as to minimize the sum of the within-group variances does not always occur at the mean. Strong unimodality is shown to be a sufficient condition for the optimal cutpoint to occur at the mean.
(1994). Principal Points of Symmetric, Strongly Unimodal Distributions. Statistics and Probability Letters, 20 (4), 253-257.