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Given a family C of plane convex bodies, let Ω, (C) be the set of all pairs (x, y) with the property that there exists K ¿ C such that ¿(K) = x and ¿(K) = y, where ¿(K) and ¿(K) denote the densities of the thinnest covering and the densest packing of the plane with copies of K, respectively. The set Ω(C) is defined analogously, with the difference that we restrict our attention to lattice packings and coverings.We prove that, for every centrally symmetric plane convex body K, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk% HiTiabes7aKnaaBaaaleaaieaacaWFmbaabeaakiaacIcacaWGlbGa% aiykaiabgsMiJkabeg9aknaaBaaaleaacaWFmbaabeaakiaacIcaca% WGlbGaaiykaiabgkHiTiaaigdacqGHKjYOcaaIXaGaaiOlaiaaikda% caaI1aWaaOaaaeaacaaIXaGaeyOeI0IaeqiTdq2aaSbaaSqaaiaa-X% eaaeqaaOGaaiikaiaadUeacaGGPaaaleq!4FD1! $$1 - \\delta _L (K) \\leqslant \\vartheta _L (K) - 1 \\leqslant 1.25\\sqrt {1 - \\delta _L (K)} $$ and give an exact analytic description of ΩL(P 8) where P 8 is the family of all centrally symmetric octagons. This allows us to show that the above inequalities are asymptotically tight.

Inequalities between lattice packing and covering densities of centrally symmetric plane convex bodies