Let $U_1,\ldots,U_n$ be independent random vectors which are uniformly distributed on the unit sphere. The random hyperplanes $U_1^\perp,\ldots,U_n^\perp$ dissect the space into a collection of random cones. A uniform random cone $S_n$ from this collection is called the Schläfli random cone. In a classical paper of Cover and Efron (1967) it was proved that the expected number of $k$-dimensional faces of a cross section of $S_n$ converges to the number of $k$-dimensional faces of a cube. We investigate the question whether a similar convergence is true also for the random shape of these cross sections. This talk is based on joint work with Zachary Kabluchko and Daniel Temesvari.

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Host: Paata Ivanisvili  pivanis@ncsu.edu