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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2009 Volume 372, Pages 108–118 (Mi znsl3563)

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

V. V. Makeev

Saint-Petersburg State University, Saint-Petersburg, Russia

Abstract: 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 $(12\sqrt2-8)S(K)/7<1.282\,S(K)$.
Theorem 3. Each three-dimensional convex body $K$ is contained in a cylinder with centrally symmetric convex base and with volume at most $(36\sqrt2-24)V(K)/7<3.845\,V(K)$. (Here, $V(K)$ denotes the volume of $K$.)
Bibl. – 5 titles.

Key words and phrases: affine regular octagon.

UDC: 514.172

Received: 25.12.2007


 English version:
Journal of Mathematical Sciences (New York), 2011, 175:5, 562–568

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© Steklov Math. Inst. of RAS, 2025