LAUSR

In the last several decades, convex geometry methods have proven very useful in algebraic geometry specifically to understand discrete invariants of algebraic varieties. An approach to study algebraic varieties is to assign to a family of varieties a corresponding family of combinatorial objects which encode geometric information about the varieties. Often, the combinatorial objects that arise are convex polytopes, and convex geometry has been an essential tool for this strategy. The emergence of convexity in algebraic geometry is rooted in the following geometric observations. Let A ⊂ Zn be a finite set, then: 1. For any vector ξ ∈ Rn, themaximum/minimumof the dot products ξ ⋅ x, x ∈ A, is attained on the boundary of the convex hull of A. 2. As k → ∞ the rescaled k-fold sums { 1 k (x1 +⋯+ xk) ∣ xi ∈ A} converge to the convex hull of A.