Self-packing of centrally symmetric convex bodies in ℝ2

AbstractLetB be a compact convex body symmetric around0 in ℝ2 which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusρ(m,B) is the smallestt such thattB can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusρ(m,B)=1+2/α(m,B) whereα(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showρ(6,B)=ρ(7,B)=3 for allB, and determine most of the largest and smallest values ofρ(m,B) form≤7. For allm we have % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaam% aalaaabaGaamyBaaqaaiabes7aKjaacIcaieqacaWFcbGaaiykaaaa% aiaawIcacaGLPaaadaahaaWcbeqaamaalyaabaGaaGymaaqaaiaaik% daaaaaaOGaeyOeI0YaaSaaaeaacaaIZaaabaGaaGOmaaaacqGHKjYO% cqaHbpGCcaGGOaGaamyBaiaacYcacaWFcbGaaiykaiabgsMiJoaabm% aabaWaaSaaaeaacaWGTbaabaGaeqiTdqMaaiikaiaa-jeacaGGPaaa% aaGaayjkaiaawMcaamaaCaaaleqabaWaaSGbaeaacaaIXaaabaGaaG% OmaaaaaaGccqGHRaWkcaaIXaGaaiilaaaa!576F! $$\left( {\frac{m}{{\delta (B)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} - \frac{3}{2} \leqslant \rho (m,B) \leqslant \left( {\frac{m}{{\delta (B)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 1,$$ whereδ(B) is the packing density ofB in ℝ2.