Holmes-Thompson area of inscribed polygons and convex projective structures

TL;DR

This paper explores the relationship between Holmes-Thompson area of inscribed polygons and ratios derived from positive flag tuples in convex projective geometry, focusing on hyperbolic quadrilaterals and thrice-punctured spheres.

Contribution

It introduces new connections between Holmes-Thompson area and flag ratios in convex projective structures, specifically for positive triples and quadruples of flags.

Findings

Holmes-Thompson area relates to double and triple ratios of flags.

Characterization of finite area convex structures on a thrice-punctured sphere.

Analysis of hyperbolic quadrilaterals within convex projective geometry.

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.