Abstract
A double-normal pair of a finite set of points that spans is a pair of points from such that lies in the closed strip bounded by the hyperplanes through and perpendicular to . A double-normal pair is strict if lies in the open strip. The problem of estimating the maximum number of double-normal pairs in a set of points in , was initiated by Martini and Soltan [Discrete Math. 290 (2005), 221–228]. It was shown in a companion paper that in the plane, this maximum is , for every . For , it follows from the Erdős–Stone theorem in extremal graph theory that for a suitable positive integer . Here we prove that and, in general, . Moreover, asymptotically we have . The same bounds hold for the maximum number of strict double-normal pairs.
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