  The last few years have seen a surge of work on high dimensional statistics
under privacy constraints, mostly following two main lines of work: the ``worst
case'' line, which does not make any distributional assumptions on the input
data; and the ``strong assumptions'' line, which assumes that the data is
generated from specific families, e.g., subgaussian distributions. In this work
we take a middle ground, obtaining new differentially private algorithms with
polynomial sample complexity for estimating quantiles in high-dimensions, as
well as estimating and sampling points of high Tukey depth, all working under
very mild distributional assumptions. From the technical perspective, our work
relies 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 notion
going back to Archimedes, which is a robust and well studied high-dimensional
analogue of the interquantile range. We show how one can privately, and with
polynomially many samples, (a) output an approximate interior point of the FB
-- e.g., ``a typical user'' in a high-dimensional database -- by leveraging the
robustness of the Steiner point of the FB; and at the expense of polynomially
many more samples, (b) produce an approximate uniform sample from the FB, by
constructing a private noisy projection oracle.
