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Abstract  A  double‐normal pair  of a finite set   S   of points that spans    R  d    is a pair of points    {  p  ,  q  }    from   S   such that   S   lies in the closed strip bounded by the hyperplanes through   p   and   q   perpendicular to    p  q   . A double‐normal pair    {  p  ,  q  }    is  strict  if   S  ∖  {  p  ,  q  }    lies in the open strip. The problem of estimating the maximum numberN  d    (  n  )of double‐normal pairs in a set of   n   points in    R  d   , 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    3  ⌊  n  /  2  ⌋   , for every    n  >  2   . For    d  ⩾  3   , it follows from the Erdős–Stone theorem in extremal graph theory thatN  d    (  n  )   =   1  2    (  1  ‐  1  /  k  )    n  2   +  o   (   n  2   )for a suitable positive integer    k  =  k  (  d  )   . Here we prove that    k  (  3  )  =  2    and, in general,    ⌈  d  /  2  ⌉  ⩽  k  (  d  )  ⩽  d  ‐  1   . Moreover, asymptotically we havelim   n  →  ∞    k   (  d  )   /  d  =  1   . The same bounds hold for the maximum number of strict double‐normal pairs.

DOUBLE‐NORMAL PAIRS IN SPACE