Holmes-Thompson area of inscribed polygons and convex projective structures

Abstract

Positive tuples of complete flags in $\mathbb{R}^3$ define two convex polygons in $\mathbb{RP}^2$, one inscribed in the other. We are interested in relating the Holmes-Thompson area of the inner polygon for the Hilbert metric on the outer polygon to the double and triple ratios of the positive tuple of flags. This article focuses on positive triples and quadruples of flags. For quadruples, we investigate the special cases of hyperbolic quadrilaterals and the parametrization of the finite area convex real projective structures on a thrice-punctured sphere.