A generalization of a result of Dembowski and Wagner

The famous Dembowski-Wagner theorem gives various characterizations of the classical geometric 2-design PG _n-1(n, q) among all 2-designs with the same parameters. One of the characterizations requires that all lines have size q + 1. It was conjectured 2] that this is also true for the designs PG _d (n, q) with 2 ≤ d ≤  n − 1. We establish this conjecture, hereby improving various previous results.