2021/09/21 by Codenotti, Giulia, Hall, Thomas, Hofscheier, Johannes · 1 citation
#52B20 (Primary) 53D05 (Secondary) #52C07 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2110.02770
Let A ∈ \ ℤ, ℝ \ and X ⊂ ℝd be a bounded set. Affine transformations given by an automorphism of ℤd and a translation in Ad are called (affine) A-unimodular transformations. The image of X under such a transformation is called an A-unimodular copy of X. It was shown in [Averkov, Hofscheier, Nill, 2019] that every convex body whose width is "big enough" contains an A-unimodular copy of X. The threshold when this happens is called the generalised flatness constant FltdA(X). It resembles the classical flatness constant if A=ℤ and X is a lattice point. In this work, we introduce a general framework for the explicit computation of these numerical constants. The approach relies on the study of A-X-free convex bodies generalising lattice-free (also known as hollow) convex bodies. We then focus on the case that X=P is a full-dimensional polytope and show that inclusion-maximal A-P-free convex bodies are polytopes. The study of those inclusion-maximal polytopes provides us with the means to explicitly determine generalised flatness constants. We apply our approach to the case X=Δ2 the standard simplex in ℝ2 of normalised volume 1 and compute Fltℝ2(Δ2)=2 and Fltℤ2(Δ2)=\frac103.