Let Me Tell You My Favorite Lattice-point Problem. . .
Integer Points in Polyhedra—Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics
This collection was compiled by Bruce Reznick from problems presented at the 2006 AMS/IMS/SIAM Summer Research Conference on Integer points in polytopes. SupposeP Rd is a convex rational d-polyhedron. The solid angle !P(x) of a point x (with respect toP) is a real number equal to the proportion of a small ball centered at x that is contained inP. That is, we let B (x) denote the ball of radius centered at x and dene !P(x) := vol (B (x)\P) volB (x) for all positive suciently small. We note that when x = 2P, !P(x) = 0; when x2P , !P(x) = 1; when x2 @P, 0 < !P(x) < 1. We dene
Beck, Matthias; Nill, Benjamin; Reznick, Bruce; Savage, Carla; Soprunov, Ivan; and Xu, Zhiqiang, "Let Me Tell You My Favorite Lattice-point Problem. . ." (2018). Mathematics Faculty Publications. 281.