In instances of medical decision-making, oftentimes there is a need to simultaneously construct statistical confidence regions for several parameters of interest. For example, there may be a need to construct confidence regions for several medians. A failure to take into account the multiplicity aspect of this problem may lead to erroneous and non-reproducible results and decisions, and non-optimal multiple confidence regions will lead to waste of information and inefficiency. Thus, it is imperative that optimal regions (with respect to a global measure of size of the regions) that satisfy a global confidence level be constructed. This talk will address this important topic. It will utilize notions of group invariance to develop optimal regions and it will then consider the issue of how to allocate a global confidence level to each of the individual problems in order to achieve optimality of the confidence regions. This will be demonstrated by considering the problem of building nonparametric confidence regions for multiple medians.