Abstract
Given a setS ofn points inRd, a subsetX of sized is called ak-simplex if the hyperplane aff(X) has exactlyk points on one side. We studyE d (k,n), the expected number of k-simplices whenS is a random sample ofn points from a probability distributionP onRd. WhenP is spherically symmetric we prove thatE d (k, n)≤cnd−1 WhenP is uniform on a convex bodyK⊂R2 we prove thatE 2 (k, n) is asymptotically linear in the rangecn≤k≤n/2 and whenk is constant it is asymptotically the expected number of vertices on the convex hull ofS. Finally, we construct a distributionP onR2 for whichE 2((n−2)/2,n) iscn logn.
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The authors express gratitude to the NSF DIMACS Center at Rutgers and Princeton. The research of I. Bárány was supported in part by Hungarian National Science Foundation Grants 1907 and 1909, and W. Steiger's research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.
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Bárány, I., Steiger, W. On the expected number of k-sets. Discrete Comput Geom 11, 243–263 (1994). https://doi.org/10.1007/BF02574008
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DOI: https://doi.org/10.1007/BF02574008