The concept of geometric quantiles and cdf is one of the most popular approaches to extend univariate quantiles and cdf to a multivariate setting. After introducing this notion, the presentation will focus on some progress made recently in our understanding of this object. Despite the fact that geometric quantile regions do not match their probability content, we show that their robustness properties, in terms of breakdown, are optimal in the sense that they correspond to their univariate antecedents. We balance this result with a less desirable property of extreme geometric quantiles, established by Girard and Stupler in 2017. We then quickly investigate the recovery of a probability measure from its geometric cdf, taking the form of a PDE and resulting in substantial different behaviors in odd and even dimensions. Finally, we will show how the geometric cdf characterizes weak convergence of probability measures and provides a natural extension of Kolmogorov's distance.
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