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Let $d \geq 4$ be a natural number and let $A$ be a finite, non-empty subsetof $\mathbb{R}^d$ such that $A$ is not contained in a translate of ahyperplane. In this setting, we show that \[ |A-A| \geq \bigg(2d - 2 +\frac{1}{d-1} \bigg) |A| - O_{d}(|A|^{1- \delta}), \] for some absoluteconstant $\delta>0$ that only depends on $d$. This provides a sharp main term,consequently answering questions of Ruzsa and Stanchescu up to an$O_{d}(|A|^{1- \delta})$ error term. We also prove new lower bounds forrestricted type difference sets and asymmetric sumsets in $\mathbb{R}^d$.

New lower bounds for cardinalities of higher dimensional difference sets and sumsets