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

V. V. Makeev

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 108-118

Résumé

Let $\gamma^3_2\colon E_2(\mathbb R^3)\to G_2(\mathbb R^3)$ be a tautological vector bundle over the Grassmann of 2-planes in $\mathbb R^3$, where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of $\mathbb R^3$. We say that a field of convex figures is given in $\gamma^3_2$ if in each fiber a convex figure is distinguished, which continuously depends on the fiber. Theorem 1. Each field of convex figures in $\gamma^3_2$ contains a figure $K$ containing a centrally symmetric convex figure with area at least $(4+16\sqrt2)S(K)/31>0.858\,S(K)$. (Here, $S(K)$ denotes the area of $K$.) Theorem 2. Each field of convex figures in $\gamma^3_2$ contains a figure $K$ that is contained in a centrally symmetric convex figure with area at most $(12\sqrt2-8)S(K)/71.282\,S(K)$. Theorem 3. Each three-dimensional convex body $K$ is contained in a cylinder with centrally symmetric convex base and with volume at most $(36\sqrt2-24)V(K)/73.845\,V(K)$. (Here, $V(K)$ denotes the volume of $K$.) Bibl. – 5 titles.