Difference sets in higher dimensions (David Conlon, Cal Tech)

Let d >= 2 be a natural number. We determine the minimum possible size of the difference set A-A in terms of |A| for any sufficiently large finite subset A of R^d that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin. Joint work with Jeck Lim.