The fence conjecture for convex projective polygons

Abstract

We prove the fence conjecture of Dumas and Wolf: for a convex projective polygon with at least five vertices, every zero of the Pick differential lies in the interior of the region obtained by cutting off the triangles at consecutive triples of vertices. The proof gives a stronger estimate for arbitrary properly convex domains. In affine-sphere coordinates normalized at a Pick zero, the domain contains the closed disk of radius $2^{-1/4}$ and its closure lies in the open disk of radius $2^{1/4}$, unless it is a rotated square. The proof compares the canonical Monge-Ampère potential of the domain with that of a normalized square. Their third-order contact gives alternating local sign sectors. A planar separation argument and the maximum principle show that the negative sectors belong to four distinct components, each meeting a different side of the square. Rotation gives the inner disk bound, and conormal duality gives the outer bound. Both constants are optimal for the class of properly convex domains normalized at a Pick zero. The argument requires neither boundary smoothness nor any restriction on the multiplicity of the Pick zero.