Journal of Mathematical Sciences

, Volume 175, Issue 5, pp 572–573 | Cite as

Affine cross-polytopes inscribed in a convex body

  • V. V. Makeev

Let X be an affine cross-polytope, i.e., the convex hull of n segments A 1 B 1,…, A n B n in \( {\mathbb{R}^n} \) that have a common midpoint O and do not lie in a hyperplane. The affine flag F(X) of X is the chain OL 1 ⊂⋯ ⊂ L n = \( {\mathbb{R}^n} \), where L k is the k-dimensional affine hull of the segments A 1 B 1,…, A k B k , kn. It is proved that each convex body K\( {\mathbb{R}^n} \) is circumscribed about an affine cross-polytope X such that the flag F(X) satisfies the following condition for each k ∈{2,…, n}:the (k−1)-planes of support at A k and B k to the body L k K in the k-plane L k are parallel to L k −1.Each such X has volume at least V(K)/2n(n−1)/2. Bibliography: 5 titles.


Russia Hull Convex Hull Convex Body Affine Hull 
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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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