2015/04/02 by Giancarlo Travaglini, Travaglini, Giancarlo, Maria Rosaria Tupputi +1 · 1 citation
Materials Science · Mathematics · #11K38 #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Point processes and geometric inequalities #Radiation Shielding Materials Analysis
paper · pdf · doi:10.48550/arxiv.1504.03251
openalex publication_date 2015/04/02 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Let C be a convex d-dimensional body. If ρ is a large positive number, then the dilated body ρC contains ρd\vert C\vert +O( ρd-1) integer points, where \vert C\vert denotes the volume of C. The above error estimate O( ρd-1) can be improved in several cases. We are interested in the L2-discrepancy DC(ρ) of a copy of ρC thrown at random in ℝd. More precisely, we consider DC(ρ):=\ ∫_\mathbbTd∫SO(d)\vert \textrmcard( ( ρσ(C)+t) ∩ℤd) - ρd\vert C\vert \vert 2dσdt\ 1/2 , where \mathbbTd= ℝd/ℤd is the d-dimensional flat torus and SO( d) is the special orthogonal group of real orthogonal matrices of determinant 1. An argument of D. Kendall shows that DC(ρ)≤ c ρ(d-1)/2. If C also satisfies the reverse inequality DC(ρ)≥ c1 ρ(d-1)/2, we say that C is L2-regular. L. Parnovski and A. Sobolev proved that, if d>1, a d-dimensional unit ball is L2% -regular if and only if d\not ≡ 1 (mod4). In this paper we characterize the L2-regular convex polygons. More precisely we prove that a convex polygon is not L2-regular if and only if it can be inscribed in a circle and it is symmetric about the centre.