The limit of finite sample breakdown point of Tukey's halfspace median for general data

Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey's halfspace median ($\frac{1} {3}$) has been obtained in literature. In this paper, we establish the result under \emph{weaker assumption} imposed on underlying distribution (halfspace symmetry) and on data set (not necessary in general position). The representation of Tukey's sample depth regions for data set \emph{not necessary in general position} is also obtained, as a by-product of our derivation.