Abstract
Tukey (1975) proposed the halfspace depth concept as a geometrical tool to handle measures. However, only recently (Koshevoy, 1999b; Struyf and Rousseeuw, 1999), it was shown that, for the class of atomic measures, this depth determines the measure. Here we extend this characterization result for the class of absolutely continuous measures for which the function exp( ) is integrable with any \( p \in {{\Bbb R}^d}\). Three issues play a key role in proving this characterization. The first, the Tukey median has a depth \( \geqslant 1/(K + 1)\) for any k-variate distribution. The second, let two measures μ and v have the same Tukey depth. Then the restrictions of these measures to any trimmed region are measures with identical Tukey depths. The third, a relation between Tukey depth of a measure with compact support and some projections of lift-zonoid. This relation allows to use the support theorem for the Radon transform.
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