Dimension bounds for finite-volume hyperbolic Coxeter polytopes

Abstract

We prove that a finite-volume hyperbolic Coxeter polytope has dimension at most $81$, substantially improving Prokhorov's bound of $995$. A refinement of Prokhorov's method gives the intermediate bound $162$, while an extension of Vinberg's planar-angle comparison to the finite-volume setting yields the sharper bound. Both arguments combine improved weighting functions with estimates using cusp geometry to control weight contributions of angles at ideal vertices or edges incident to them.