Berman-Hartmanis Isomorphism for NP$_{\mathbb{R}}$?
Depending on which version of $\mathsf{NP}_{\mathbb{R}}$ you use, yes or it's open. When one considers BSS machines that only use addition and subtaction, and only branch on equality, the answer is yes. If one includes branching on $&lt;$, I believe it is still open, and the same if one allows multiplications. For details, see Cucker, Koiran, and Matamala "Complexity and dimension," Inform. Proc. Lett. 1997