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Journal of Mathematical Sciences

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

Affine cross-polytopes inscribed in a convex body

  • V. V. Makeev
Article
  • 21 Downloads

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.

Keywords

Russia Hull Convex Hull Convex Body Affine Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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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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