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In this article, we prove that the union of two almost orthogonal planes in ℝ 4  is Almgren‐minimal. This gives an example of a one‐parameter family of minimal cones, which is a phenomenon that does not exist in ℝ 3 . This work is motivated by an attempt to classify the singularities of 2‐dimensional Almgren‐minimal sets in ℝ 4 . Note that the traditional methods for proving minimality (calibrations and slicing arguments) do not apply here, we are obliged to use some more complicated arguments such as a stopping time argument, harmonic extensions, Federer–Fleming projections, etc., which are rarely used to prove minimality (rather they are often used to prove regularity). The regularity results for 2‐dimensional Almgren minimal sets [G. David, ‘Hölder regularity of two‐dimensional almost‐minimal sets in ℝ  n  ’,  Ann. Fac. Sci. Toulouse  XVIII (2009) 65–246; G. David, ‘C 1+α ‐regularity for two‐dimensional almost‐minimal sets in ℝ  n  .’  J. Geom. Anal.  20 (2010) 837–954] are also needed here.

Almgren‐minimality of unions of two almost orthogonal planes in ℝ 4