Given a set of points in R , two points , from form a double-normal pair, if the set lies between two parallel hyperplanes that pass through and , respectively, and that are orthogonal to the segment . In this paper we study the maximum number ( ) of double-normal pairs in a set of points in R . It is not difficult to get from the famous Erdős-Stone theorem that ( ) = 1 2 (1 − 1/ ) 2 + ( 2 ) for a suitable integer = ( ) and it was shown in the paper by J. Pach and K. Swanepoel that ⌈ /2⌉ ≤ ( ) ≤ − 1 and that asymptotically ( ) − (log ). In this paper we sharpen the upper bound on ( ), which, in particular, gives (4) = 2 and (5) = 3 in addition to the equality (3) = 2 established by J. Pach and K. Swanepoel. Asymptotically we get ( ) ≤ − log 2 ( ) = − (1 + (1)) log 2 ( ) and show that this problem is connected with the problem of determining the maximum number of points in R that form pairwise acute (or non-obtuse) angles.