Projective Invariants and Hilbert Geometry on Convex Polytopes — E8 Intelligence Research

FINDING: The cross-ratio is the fundamental projective invariant linking collinearity, concurrency, and harmonic ranges; Hilbert geometry on convex polytopes is characterized by volume growth and bi-Lipschitz equivalence to simplex geometry. | MATH: Cross-ratio for four collinear points \( (A,B;C,D) = \frac{AC/BC}{AD/BD} \); harmonic range when cross-ratio = −1 (or 1 depending on convention), i.e., \( (A,B;C,D) = -1 \) implies \( C,D \) are harmonic conjugates w.r.t. \( A,B \). Hilbert metric on convex domain \( \Omega \): \( d_H(x,y) = \frac{1}{2} \ln \left( \frac{|a-y||b-x|}{|a-x||b-y|} \right) \) where \( a,b \) are boundary intersections of line through \( x,y \). For polytopes, volume growth entropy \( h = \lim_{R\to\infty} \frac{1}{R} \ln \text{Vol}(B(x,R)) \) is finite and equals a combinatorial invariant; bi-Lipschitz equivalence to simplex geometry means the polytope Hilbert geometry is quasi-isometric to the simplex's. | CONNECTION: The cross-ratio's special value −1 (harmoni Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com