Minimality of planes in normed spaces

We prove that a region in a two-dimensional affine subspace of a normed space $V$ has the least 2-dimensional Hausdorff measure among all compact surfaces with the same boundary. Furthermore, the 2-dimensional Hausdorff area density admits a convex extension to $Λ^2 V$. The proof is based on a (probably) new inequality for the Euclidean area of a convex centrally-symmetric polygon.

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