  Let $d \geq 2$ be a natural number. We show that $$|A-A| \geq \left(2d-2 +
\frac{1}{d-1}\right)|A|-(2d^2-4d+3)$$ for any sufficiently large finite subset
$A$ of $\mathbb{R}^d$ 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.
