On sets defining few ordinary solids

2018/08/20 by Ball, Simeon, Jimenez, Enrique
#51M04 #52C35 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1808.06388

Abstract

Let S be a set of n points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of S is less than Kn³ for some K=o(n^(1)/(7)) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size n of an elliptic curve which span less than (1)/(6)n³ solids containing exactly four points of S.

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