Half space depth and regression depth characterize the empirical distribution

For multivariate data, the halfspace depth function can be seen as a natura l and affine equivariant generalization of the univariate empirical cdf. Fo r any multivariate data set, we show that the resulting halfspace depth Fun ction completely determines the empirical distribution. We do this by actua lly reconstructing the data points from the depth contours. The data need n ot be in general position. Moreover, we prove the same property for regress ion depth. (C) 1999 Academic Press.