Test Functions, Partitioning and Concentration Phenomena in Toric Geometry

We discuss simple partitioning phenomena for intersection rings of certain toric varieties, and apply this to prove the following results. ̋ We generalize a result of Gräbe, proving that Stanley–Reisner rings of homology spheres are Gorenstein. ̋ We give a new and direct proof of a theorem of the authors due to which the cohomology rings of matroids are Poincare duality algebras. ̋ We prove a conjecture of Zharkov relating partitioning to Orlik–Solomon algebras of fans and matroids. ̋ We characterize simplicial polytopes with extremal primitive Betti numbers, and apply this to resolve a conjecture of Kalai concerning polytopes approximating smooth convex bodies.