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The last few years have seen a surge of work on high dimensional statisticsunder privacy constraints, mostly following two main lines of work: the ``worstcase'' line, which does not make any distributional assumptions on the inputdata; and the ``strong assumptions'' line, which assumes that the data isgenerated from specific families, e.g., subgaussian distributions. In this workwe take a middle ground, obtaining new differentially private algorithms withpolynomial sample complexity for estimating quantiles in high-dimensions, aswell as estimating and sampling points of high Tukey depth, all working undervery mild distributional assumptions. From the technical perspective, our workrelies upon deep robustness results in the convex geometry literature,demonstrating how such results can be used in a private context.  Our main object of interest is the (convex) floating body (FB), a notiongoing back to Archimedes, which is a robust and well studied high-dimensionalanalogue of the interquantile range. We show how one can privately, and withpolynomially many samples, (a) output an approximate interior point of the FB-- e.g., ``a typical user'' in a high-dimensional database -- by leveraging therobustness of the Steiner point of the FB; and at the expense of polynomiallymany more samples, (b) produce an approximate uniform sample from the FB, byconstructing a private noisy projection oracle.

Archimedes Meets Privacy: On Privately Estimating Quantiles in High Dimensions Under Minimal Assumptions