Fleming (1962) gave a new proof of Bernstein's theorem by deducing it from the fact that there is no non-planar area-minimizing cone in R3.
Almgren (1966) showed there are no non-planar minimizing cones in R4, thus extending Bernstein's theorem to R5.
Simons (1968) showed there are no non-planar minimizing cones in R7, thus extending Bernstein's theorem to R8.
He also showed that the surface defined by is a locally stable cone in R8, and asked if it is globally area-minimizing.
Combined with the result of Simons, this shows that the analogue of Bernstein's theorem is true in Rn for n≤8, and false in higher dimensions.