Point-balanced arrangements in the real projective plane and the Hirzebruch property

A collection of $m\ge 2$ lines through the origin in ${\mathbb{R}}^2$ is balanced if the sum of the orthogonal projections onto these lines equals $m\mathrm{/}2$ times the identity. A line arrangement in ${\mathbb{R}\mathbb{P}}^2$, endowed with the round metric of curvature $1$, is point-balanced if the tangent lines at every vertex form a balanced collection. A line arrangement $\mathcal{A}$ in ${\mathbb{R}\mathbb{P}}^2$ has the Hirzebruch property if it consists of $3k$ lines and every line contains exactly $k+1$ vertices. Using a Kempf–Ness convexity argument, we show that an irreducible arrangement $\mathcal{A}$ has the Hirzebruch property if and only if its projective equivalence class contains a point-balanced representative, unique up to orthogonal transformations. We then combine elementary properties of balanced collections of lines in ${\mathbb{R}}^2$ with spherical geometry to prove that every irreducible point-balanced arrangement is a reflection arrangement. This gives a new proof of Panov’s classification of real Hirzebruch arrangements.