Region of maximum probability content of fixed diameter

In this article it is shown that when the pdff(x) is non-increasing in xi, i = 1, 2, ..., k, the sphere of diameter d centred at the origin has largest probability content as compared to any other region of diameter at most d. As a consequence, it follows that a sphere of diameter d has largest volume among the regions of diameter at most d. Further, the results are used to obtain certain optimum statistical decision rules.