Measure of central symmetry for fields of convex figures and three-dimensional convex bodies

Abstract

Let $ \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) $ be the tautological vector bundle over the Grassmann manifold of 2-planes in $ {\mathbb{R}^3} $ , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of $ {\mathbb{R}^3} $ . A field of convex figures is given in γ 2 3 if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ 2 3 contains a figure K containing a centrally symmetric convex figure of area $ \left( {4 + 16\sqrt {2} } \right) $ S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area $ \left( {12\sqrt {2} - 8} \right) $ S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume $ \left( {36\sqrt {2} - 24} \right) $ V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.