Minimality of planes in normed spaces

TL;DR

This paper proves that planes in normed spaces minimize 2D Hausdorff measure among surfaces with the same boundary and introduces a convex extension for area density, based on a new inequality for convex polygons.

Contribution

It establishes minimality of planes in normed spaces for 2D Hausdorff measure and introduces a convex extension of area density, along with a new polygon inequality.

Findings

Planes minimize 2D Hausdorff measure in normed spaces.

A convex extension of 2D Hausdorff area density is constructed.

A new inequality for convex centrally-symmetric polygons is developed.

Abstract

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 $\Lambda^2 V$. The proof is based on a (probably) new inequality for the Euclidean area of a convex centrally-symmetric polygon.